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3564 lines
142 KiB
3564 lines
142 KiB
5 months ago
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r"""
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This module contains :py:meth:`~sympy.solvers.ode.dsolve` and different helper
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functions that it uses.
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:py:meth:`~sympy.solvers.ode.dsolve` solves ordinary differential equations.
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See the docstring on the various functions for their uses. Note that partial
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differential equations support is in ``pde.py``. Note that hint functions
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have docstrings describing their various methods, but they are intended for
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internal use. Use ``dsolve(ode, func, hint=hint)`` to solve an ODE using a
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specific hint. See also the docstring on
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:py:meth:`~sympy.solvers.ode.dsolve`.
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**Functions in this module**
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These are the user functions in this module:
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- :py:meth:`~sympy.solvers.ode.dsolve` - Solves ODEs.
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- :py:meth:`~sympy.solvers.ode.classify_ode` - Classifies ODEs into
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possible hints for :py:meth:`~sympy.solvers.ode.dsolve`.
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- :py:meth:`~sympy.solvers.ode.checkodesol` - Checks if an equation is the
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solution to an ODE.
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- :py:meth:`~sympy.solvers.ode.homogeneous_order` - Returns the
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homogeneous order of an expression.
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- :py:meth:`~sympy.solvers.ode.infinitesimals` - Returns the infinitesimals
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of the Lie group of point transformations of an ODE, such that it is
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invariant.
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- :py:meth:`~sympy.solvers.ode.checkinfsol` - Checks if the given infinitesimals
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are the actual infinitesimals of a first order ODE.
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These are the non-solver helper functions that are for internal use. The
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user should use the various options to
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:py:meth:`~sympy.solvers.ode.dsolve` to obtain the functionality provided
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by these functions:
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- :py:meth:`~sympy.solvers.ode.ode.odesimp` - Does all forms of ODE
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simplification.
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- :py:meth:`~sympy.solvers.ode.ode.ode_sol_simplicity` - A key function for
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comparing solutions by simplicity.
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- :py:meth:`~sympy.solvers.ode.constantsimp` - Simplifies arbitrary
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constants.
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- :py:meth:`~sympy.solvers.ode.ode.constant_renumber` - Renumber arbitrary
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constants.
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- :py:meth:`~sympy.solvers.ode.ode._handle_Integral` - Evaluate unevaluated
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Integrals.
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See also the docstrings of these functions.
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**Currently implemented solver methods**
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The following methods are implemented for solving ordinary differential
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equations. See the docstrings of the various hint functions for more
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information on each (run ``help(ode)``):
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- 1st order separable differential equations.
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- 1st order differential equations whose coefficients or `dx` and `dy` are
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functions homogeneous of the same order.
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- 1st order exact differential equations.
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- 1st order linear differential equations.
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- 1st order Bernoulli differential equations.
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- Power series solutions for first order differential equations.
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- Lie Group method of solving first order differential equations.
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- 2nd order Liouville differential equations.
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- Power series solutions for second order differential equations
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at ordinary and regular singular points.
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- `n`\th order differential equation that can be solved with algebraic
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rearrangement and integration.
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- `n`\th order linear homogeneous differential equation with constant
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coefficients.
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- `n`\th order linear inhomogeneous differential equation with constant
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coefficients using the method of undetermined coefficients.
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- `n`\th order linear inhomogeneous differential equation with constant
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coefficients using the method of variation of parameters.
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**Philosophy behind this module**
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This module is designed to make it easy to add new ODE solving methods without
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having to mess with the solving code for other methods. The idea is that
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there is a :py:meth:`~sympy.solvers.ode.classify_ode` function, which takes in
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an ODE and tells you what hints, if any, will solve the ODE. It does this
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without attempting to solve the ODE, so it is fast. Each solving method is a
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hint, and it has its own function, named ``ode_<hint>``. That function takes
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in the ODE and any match expression gathered by
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:py:meth:`~sympy.solvers.ode.classify_ode` and returns a solved result. If
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this result has any integrals in it, the hint function will return an
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unevaluated :py:class:`~sympy.integrals.integrals.Integral` class.
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:py:meth:`~sympy.solvers.ode.dsolve`, which is the user wrapper function
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around all of this, will then call :py:meth:`~sympy.solvers.ode.ode.odesimp` on
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the result, which, among other things, will attempt to solve the equation for
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the dependent variable (the function we are solving for), simplify the
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arbitrary constants in the expression, and evaluate any integrals, if the hint
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allows it.
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**How to add new solution methods**
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If you have an ODE that you want :py:meth:`~sympy.solvers.ode.dsolve` to be
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able to solve, try to avoid adding special case code here. Instead, try
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finding a general method that will solve your ODE, as well as others. This
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way, the :py:mod:`~sympy.solvers.ode` module will become more robust, and
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unhindered by special case hacks. WolphramAlpha and Maple's
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DETools[odeadvisor] function are two resources you can use to classify a
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specific ODE. It is also better for a method to work with an `n`\th order ODE
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instead of only with specific orders, if possible.
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To add a new method, there are a few things that you need to do. First, you
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need a hint name for your method. Try to name your hint so that it is
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unambiguous with all other methods, including ones that may not be implemented
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yet. If your method uses integrals, also include a ``hint_Integral`` hint.
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If there is more than one way to solve ODEs with your method, include a hint
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for each one, as well as a ``<hint>_best`` hint. Your ``ode_<hint>_best()``
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function should choose the best using min with ``ode_sol_simplicity`` as the
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key argument. See
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:obj:`~sympy.solvers.ode.single.HomogeneousCoeffBest`, for example.
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The function that uses your method will be called ``ode_<hint>()``, so the
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hint must only use characters that are allowed in a Python function name
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(alphanumeric characters and the underscore '``_``' character). Include a
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function for every hint, except for ``_Integral`` hints
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(:py:meth:`~sympy.solvers.ode.dsolve` takes care of those automatically).
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Hint names should be all lowercase, unless a word is commonly capitalized
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(such as Integral or Bernoulli). If you have a hint that you do not want to
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run with ``all_Integral`` that does not have an ``_Integral`` counterpart (such
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as a best hint that would defeat the purpose of ``all_Integral``), you will
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need to remove it manually in the :py:meth:`~sympy.solvers.ode.dsolve` code.
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See also the :py:meth:`~sympy.solvers.ode.classify_ode` docstring for
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guidelines on writing a hint name.
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Determine *in general* how the solutions returned by your method compare with
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other methods that can potentially solve the same ODEs. Then, put your hints
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in the :py:data:`~sympy.solvers.ode.allhints` tuple in the order that they
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should be called. The ordering of this tuple determines which hints are
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default. Note that exceptions are ok, because it is easy for the user to
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choose individual hints with :py:meth:`~sympy.solvers.ode.dsolve`. In
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general, ``_Integral`` variants should go at the end of the list, and
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``_best`` variants should go before the various hints they apply to. For
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example, the ``undetermined_coefficients`` hint comes before the
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``variation_of_parameters`` hint because, even though variation of parameters
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is more general than undetermined coefficients, undetermined coefficients
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generally returns cleaner results for the ODEs that it can solve than
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variation of parameters does, and it does not require integration, so it is
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much faster.
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Next, you need to have a match expression or a function that matches the type
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of the ODE, which you should put in :py:meth:`~sympy.solvers.ode.classify_ode`
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(if the match function is more than just a few lines. It should match the
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ODE without solving for it as much as possible, so that
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:py:meth:`~sympy.solvers.ode.classify_ode` remains fast and is not hindered by
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bugs in solving code. Be sure to consider corner cases. For example, if your
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solution method involves dividing by something, make sure you exclude the case
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where that division will be 0.
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In most cases, the matching of the ODE will also give you the various parts
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that you need to solve it. You should put that in a dictionary (``.match()``
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will do this for you), and add that as ``matching_hints['hint'] = matchdict``
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in the relevant part of :py:meth:`~sympy.solvers.ode.classify_ode`.
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:py:meth:`~sympy.solvers.ode.classify_ode` will then send this to
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:py:meth:`~sympy.solvers.ode.dsolve`, which will send it to your function as
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the ``match`` argument. Your function should be named ``ode_<hint>(eq, func,
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order, match)`. If you need to send more information, put it in the ``match``
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dictionary. For example, if you had to substitute in a dummy variable in
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:py:meth:`~sympy.solvers.ode.classify_ode` to match the ODE, you will need to
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pass it to your function using the `match` dict to access it. You can access
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the independent variable using ``func.args[0]``, and the dependent variable
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(the function you are trying to solve for) as ``func.func``. If, while trying
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to solve the ODE, you find that you cannot, raise ``NotImplementedError``.
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:py:meth:`~sympy.solvers.ode.dsolve` will catch this error with the ``all``
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meta-hint, rather than causing the whole routine to fail.
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Add a docstring to your function that describes the method employed. Like
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with anything else in SymPy, you will need to add a doctest to the docstring,
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in addition to real tests in ``test_ode.py``. Try to maintain consistency
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with the other hint functions' docstrings. Add your method to the list at the
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top of this docstring. Also, add your method to ``ode.rst`` in the
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``docs/src`` directory, so that the Sphinx docs will pull its docstring into
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the main SymPy documentation. Be sure to make the Sphinx documentation by
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running ``make html`` from within the doc directory to verify that the
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docstring formats correctly.
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If your solution method involves integrating, use :py:obj:`~.Integral` instead of
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:py:meth:`~sympy.core.expr.Expr.integrate`. This allows the user to bypass
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hard/slow integration by using the ``_Integral`` variant of your hint. In
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most cases, calling :py:meth:`sympy.core.basic.Basic.doit` will integrate your
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solution. If this is not the case, you will need to write special code in
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:py:meth:`~sympy.solvers.ode.ode._handle_Integral`. Arbitrary constants should be
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symbols named ``C1``, ``C2``, and so on. All solution methods should return
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an equality instance. If you need an arbitrary number of arbitrary constants,
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you can use ``constants = numbered_symbols(prefix='C', cls=Symbol, start=1)``.
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If it is possible to solve for the dependent function in a general way, do so.
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Otherwise, do as best as you can, but do not call solve in your
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``ode_<hint>()`` function. :py:meth:`~sympy.solvers.ode.ode.odesimp` will attempt
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to solve the solution for you, so you do not need to do that. Lastly, if your
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ODE has a common simplification that can be applied to your solutions, you can
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add a special case in :py:meth:`~sympy.solvers.ode.ode.odesimp` for it. For
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example, solutions returned from the ``1st_homogeneous_coeff`` hints often
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have many :obj:`~sympy.functions.elementary.exponential.log` terms, so
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:py:meth:`~sympy.solvers.ode.ode.odesimp` calls
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:py:meth:`~sympy.simplify.simplify.logcombine` on them (it also helps to write
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the arbitrary constant as ``log(C1)`` instead of ``C1`` in this case). Also
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consider common ways that you can rearrange your solution to have
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:py:meth:`~sympy.solvers.ode.constantsimp` take better advantage of it. It is
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better to put simplification in :py:meth:`~sympy.solvers.ode.ode.odesimp` than in
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your method, because it can then be turned off with the simplify flag in
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:py:meth:`~sympy.solvers.ode.dsolve`. If you have any extraneous
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simplification in your function, be sure to only run it using ``if
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match.get('simplify', True):``, especially if it can be slow or if it can
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reduce the domain of the solution.
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Finally, as with every contribution to SymPy, your method will need to be
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tested. Add a test for each method in ``test_ode.py``. Follow the
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conventions there, i.e., test the solver using ``dsolve(eq, f(x),
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hint=your_hint)``, and also test the solution using
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:py:meth:`~sympy.solvers.ode.checkodesol` (you can put these in a separate
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tests and skip/XFAIL if it runs too slow/does not work). Be sure to call your
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hint specifically in :py:meth:`~sympy.solvers.ode.dsolve`, that way the test
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will not be broken simply by the introduction of another matching hint. If your
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method works for higher order (>1) ODEs, you will need to run ``sol =
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constant_renumber(sol, 'C', 1, order)`` for each solution, where ``order`` is
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the order of the ODE. This is because ``constant_renumber`` renumbers the
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arbitrary constants by printing order, which is platform dependent. Try to
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test every corner case of your solver, including a range of orders if it is a
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`n`\th order solver, but if your solver is slow, such as if it involves hard
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integration, try to keep the test run time down.
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Feel free to refactor existing hints to avoid duplicating code or creating
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inconsistencies. If you can show that your method exactly duplicates an
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existing method, including in the simplicity and speed of obtaining the
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solutions, then you can remove the old, less general method. The existing
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code is tested extensively in ``test_ode.py``, so if anything is broken, one
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of those tests will surely fail.
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"""
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from sympy.core import Add, S, Mul, Pow, oo
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from sympy.core.containers import Tuple
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from sympy.core.expr import AtomicExpr, Expr
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from sympy.core.function import (Function, Derivative, AppliedUndef, diff,
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expand, expand_mul, Subs)
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from sympy.core.multidimensional import vectorize
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from sympy.core.numbers import nan, zoo, Number
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from sympy.core.relational import Equality, Eq
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from sympy.core.sorting import default_sort_key, ordered
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from sympy.core.symbol import Symbol, Wild, Dummy, symbols
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from sympy.core.sympify import sympify
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from sympy.core.traversal import preorder_traversal
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from sympy.logic.boolalg import (BooleanAtom, BooleanTrue,
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BooleanFalse)
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from sympy.functions import exp, log, sqrt
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from sympy.functions.combinatorial.factorials import factorial
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from sympy.integrals.integrals import Integral
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from sympy.polys import (Poly, terms_gcd, PolynomialError, lcm)
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from sympy.polys.polytools import cancel
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from sympy.series import Order
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from sympy.series.series import series
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from sympy.simplify import (collect, logcombine, powsimp, # type: ignore
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separatevars, simplify, cse)
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from sympy.simplify.radsimp import collect_const
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from sympy.solvers import checksol, solve
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from sympy.utilities import numbered_symbols
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from sympy.utilities.iterables import uniq, sift, iterable
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from sympy.solvers.deutils import _preprocess, ode_order, _desolve
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#: This is a list of hints in the order that they should be preferred by
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#: :py:meth:`~sympy.solvers.ode.classify_ode`. In general, hints earlier in the
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#: list should produce simpler solutions than those later in the list (for
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#: ODEs that fit both). For now, the order of this list is based on empirical
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#: observations by the developers of SymPy.
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#:
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#: The hint used by :py:meth:`~sympy.solvers.ode.dsolve` for a specific ODE
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#: can be overridden (see the docstring).
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#:
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#: In general, ``_Integral`` hints are grouped at the end of the list, unless
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#: there is a method that returns an unevaluable integral most of the time
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#: (which go near the end of the list anyway). ``default``, ``all``,
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#: ``best``, and ``all_Integral`` meta-hints should not be included in this
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#: list, but ``_best`` and ``_Integral`` hints should be included.
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allhints = (
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"factorable",
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"nth_algebraic",
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"separable",
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"1st_exact",
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"1st_linear",
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"Bernoulli",
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"1st_rational_riccati",
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"Riccati_special_minus2",
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"1st_homogeneous_coeff_best",
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"1st_homogeneous_coeff_subs_indep_div_dep",
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"1st_homogeneous_coeff_subs_dep_div_indep",
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|
|
"almost_linear",
|
||
|
|
"linear_coefficients",
|
||
|
|
"separable_reduced",
|
||
|
|
"1st_power_series",
|
||
|
|
"lie_group",
|
||
|
|
"nth_linear_constant_coeff_homogeneous",
|
||
|
|
"nth_linear_euler_eq_homogeneous",
|
||
|
|
"nth_linear_constant_coeff_undetermined_coefficients",
|
||
|
|
"nth_linear_euler_eq_nonhomogeneous_undetermined_coefficients",
|
||
|
|
"nth_linear_constant_coeff_variation_of_parameters",
|
||
|
|
"nth_linear_euler_eq_nonhomogeneous_variation_of_parameters",
|
||
|
|
"Liouville",
|
||
|
|
"2nd_linear_airy",
|
||
|
|
"2nd_linear_bessel",
|
||
|
|
"2nd_hypergeometric",
|
||
|
|
"2nd_hypergeometric_Integral",
|
||
|
|
"nth_order_reducible",
|
||
|
|
"2nd_power_series_ordinary",
|
||
|
|
"2nd_power_series_regular",
|
||
|
|
"nth_algebraic_Integral",
|
||
|
|
"separable_Integral",
|
||
|
|
"1st_exact_Integral",
|
||
|
|
"1st_linear_Integral",
|
||
|
|
"Bernoulli_Integral",
|
||
|
|
"1st_homogeneous_coeff_subs_indep_div_dep_Integral",
|
||
|
|
"1st_homogeneous_coeff_subs_dep_div_indep_Integral",
|
||
|
|
"almost_linear_Integral",
|
||
|
|
"linear_coefficients_Integral",
|
||
|
|
"separable_reduced_Integral",
|
||
|
|
"nth_linear_constant_coeff_variation_of_parameters_Integral",
|
||
|
|
"nth_linear_euler_eq_nonhomogeneous_variation_of_parameters_Integral",
|
||
|
|
"Liouville_Integral",
|
||
|
|
"2nd_nonlinear_autonomous_conserved",
|
||
|
|
"2nd_nonlinear_autonomous_conserved_Integral",
|
||
|
|
)
|
||
|
|
|
||
|
|
|
||
|
|
|
||
|
|
def get_numbered_constants(eq, num=1, start=1, prefix='C'):
|
||
|
|
"""
|
||
|
|
Returns a list of constants that do not occur
|
||
|
|
in eq already.
|
||
|
|
"""
|
||
|
|
|
||
|
|
ncs = iter_numbered_constants(eq, start, prefix)
|
||
|
|
Cs = [next(ncs) for i in range(num)]
|
||
|
|
return (Cs[0] if num == 1 else tuple(Cs))
|
||
|
|
|
||
|
|
|
||
|
|
def iter_numbered_constants(eq, start=1, prefix='C'):
|
||
|
|
"""
|
||
|
|
Returns an iterator of constants that do not occur
|
||
|
|
in eq already.
|
||
|
|
"""
|
||
|
|
|
||
|
|
if isinstance(eq, (Expr, Eq)):
|
||
|
|
eq = [eq]
|
||
|
|
elif not iterable(eq):
|
||
|
|
raise ValueError("Expected Expr or iterable but got %s" % eq)
|
||
|
|
|
||
|
|
atom_set = set().union(*[i.free_symbols for i in eq])
|
||
|
|
func_set = set().union(*[i.atoms(Function) for i in eq])
|
||
|
|
if func_set:
|
||
|
|
atom_set |= {Symbol(str(f.func)) for f in func_set}
|
||
|
|
return numbered_symbols(start=start, prefix=prefix, exclude=atom_set)
|
||
|
|
|
||
|
|
|
||
|
|
def dsolve(eq, func=None, hint="default", simplify=True,
|
||
|
|
ics= None, xi=None, eta=None, x0=0, n=6, **kwargs):
|
||
|
|
r"""
|
||
|
|
Solves any (supported) kind of ordinary differential equation and
|
||
|
|
system of ordinary differential equations.
|
||
|
|
|
||
|
|
For single ordinary differential equation
|
||
|
|
=========================================
|
||
|
|
|
||
|
|
It is classified under this when number of equation in ``eq`` is one.
|
||
|
|
**Usage**
|
||
|
|
|
||
|
|
``dsolve(eq, f(x), hint)`` -> Solve ordinary differential equation
|
||
|
|
``eq`` for function ``f(x)``, using method ``hint``.
|
||
|
|
|
||
|
|
**Details**
|
||
|
|
|
||
|
|
``eq`` can be any supported ordinary differential equation (see the
|
||
|
|
:py:mod:`~sympy.solvers.ode` docstring for supported methods).
|
||
|
|
This can either be an :py:class:`~sympy.core.relational.Equality`,
|
||
|
|
or an expression, which is assumed to be equal to ``0``.
|
||
|
|
|
||
|
|
``f(x)`` is a function of one variable whose derivatives in that
|
||
|
|
variable make up the ordinary differential equation ``eq``. In
|
||
|
|
many cases it is not necessary to provide this; it will be
|
||
|
|
autodetected (and an error raised if it could not be detected).
|
||
|
|
|
||
|
|
``hint`` is the solving method that you want dsolve to use. Use
|
||
|
|
``classify_ode(eq, f(x))`` to get all of the possible hints for an
|
||
|
|
ODE. The default hint, ``default``, will use whatever hint is
|
||
|
|
returned first by :py:meth:`~sympy.solvers.ode.classify_ode`. See
|
||
|
|
Hints below for more options that you can use for hint.
|
||
|
|
|
||
|
|
``simplify`` enables simplification by
|
||
|
|
:py:meth:`~sympy.solvers.ode.ode.odesimp`. See its docstring for more
|
||
|
|
information. Turn this off, for example, to disable solving of
|
||
|
|
solutions for ``func`` or simplification of arbitrary constants.
|
||
|
|
It will still integrate with this hint. Note that the solution may
|
||
|
|
contain more arbitrary constants than the order of the ODE with
|
||
|
|
this option enabled.
|
||
|
|
|
||
|
|
``xi`` and ``eta`` are the infinitesimal functions of an ordinary
|
||
|
|
differential equation. They are the infinitesimals of the Lie group
|
||
|
|
of point transformations for which the differential equation is
|
||
|
|
invariant. The user can specify values for the infinitesimals. If
|
||
|
|
nothing is specified, ``xi`` and ``eta`` are calculated using
|
||
|
|
:py:meth:`~sympy.solvers.ode.infinitesimals` with the help of various
|
||
|
|
heuristics.
|
||
|
|
|
||
|
|
``ics`` is the set of initial/boundary conditions for the differential equation.
|
||
|
|
It should be given in the form of ``{f(x0): x1, f(x).diff(x).subs(x, x2):
|
||
|
|
x3}`` and so on. For power series solutions, if no initial
|
||
|
|
conditions are specified ``f(0)`` is assumed to be ``C0`` and the power
|
||
|
|
series solution is calculated about 0.
|
||
|
|
|
||
|
|
``x0`` is the point about which the power series solution of a differential
|
||
|
|
equation is to be evaluated.
|
||
|
|
|
||
|
|
``n`` gives the exponent of the dependent variable up to which the power series
|
||
|
|
solution of a differential equation is to be evaluated.
|
||
|
|
|
||
|
|
**Hints**
|
||
|
|
|
||
|
|
Aside from the various solving methods, there are also some meta-hints
|
||
|
|
that you can pass to :py:meth:`~sympy.solvers.ode.dsolve`:
|
||
|
|
|
||
|
|
``default``:
|
||
|
|
This uses whatever hint is returned first by
|
||
|
|
:py:meth:`~sympy.solvers.ode.classify_ode`. This is the
|
||
|
|
default argument to :py:meth:`~sympy.solvers.ode.dsolve`.
|
||
|
|
|
||
|
|
``all``:
|
||
|
|
To make :py:meth:`~sympy.solvers.ode.dsolve` apply all
|
||
|
|
relevant classification hints, use ``dsolve(ODE, func,
|
||
|
|
hint="all")``. This will return a dictionary of
|
||
|
|
``hint:solution`` terms. If a hint causes dsolve to raise the
|
||
|
|
``NotImplementedError``, value of that hint's key will be the
|
||
|
|
exception object raised. The dictionary will also include
|
||
|
|
some special keys:
|
||
|
|
|
||
|
|
- ``order``: The order of the ODE. See also
|
||
|
|
:py:meth:`~sympy.solvers.deutils.ode_order` in
|
||
|
|
``deutils.py``.
|
||
|
|
- ``best``: The simplest hint; what would be returned by
|
||
|
|
``best`` below.
|
||
|
|
- ``best_hint``: The hint that would produce the solution
|
||
|
|
given by ``best``. If more than one hint produces the best
|
||
|
|
solution, the first one in the tuple returned by
|
||
|
|
:py:meth:`~sympy.solvers.ode.classify_ode` is chosen.
|
||
|
|
- ``default``: The solution that would be returned by default.
|
||
|
|
This is the one produced by the hint that appears first in
|
||
|
|
the tuple returned by
|
||
|
|
:py:meth:`~sympy.solvers.ode.classify_ode`.
|
||
|
|
|
||
|
|
``all_Integral``:
|
||
|
|
This is the same as ``all``, except if a hint also has a
|
||
|
|
corresponding ``_Integral`` hint, it only returns the
|
||
|
|
``_Integral`` hint. This is useful if ``all`` causes
|
||
|
|
:py:meth:`~sympy.solvers.ode.dsolve` to hang because of a
|
||
|
|
difficult or impossible integral. This meta-hint will also be
|
||
|
|
much faster than ``all``, because
|
||
|
|
:py:meth:`~sympy.core.expr.Expr.integrate` is an expensive
|
||
|
|
routine.
|
||
|
|
|
||
|
|
``best``:
|
||
|
|
To have :py:meth:`~sympy.solvers.ode.dsolve` try all methods
|
||
|
|
and return the simplest one. This takes into account whether
|
||
|
|
the solution is solvable in the function, whether it contains
|
||
|
|
any Integral classes (i.e. unevaluatable integrals), and
|
||
|
|
which one is the shortest in size.
|
||
|
|
|
||
|
|
See also the :py:meth:`~sympy.solvers.ode.classify_ode` docstring for
|
||
|
|
more info on hints, and the :py:mod:`~sympy.solvers.ode` docstring for
|
||
|
|
a list of all supported hints.
|
||
|
|
|
||
|
|
**Tips**
|
||
|
|
|
||
|
|
- You can declare the derivative of an unknown function this way:
|
||
|
|
|
||
|
|
>>> from sympy import Function, Derivative
|
||
|
|
>>> from sympy.abc import x # x is the independent variable
|
||
|
|
>>> f = Function("f")(x) # f is a function of x
|
||
|
|
>>> # f_ will be the derivative of f with respect to x
|
||
|
|
>>> f_ = Derivative(f, x)
|
||
|
|
|
||
|
|
- See ``test_ode.py`` for many tests, which serves also as a set of
|
||
|
|
examples for how to use :py:meth:`~sympy.solvers.ode.dsolve`.
|
||
|
|
- :py:meth:`~sympy.solvers.ode.dsolve` always returns an
|
||
|
|
:py:class:`~sympy.core.relational.Equality` class (except for the
|
||
|
|
case when the hint is ``all`` or ``all_Integral``). If possible, it
|
||
|
|
solves the solution explicitly for the function being solved for.
|
||
|
|
Otherwise, it returns an implicit solution.
|
||
|
|
- Arbitrary constants are symbols named ``C1``, ``C2``, and so on.
|
||
|
|
- Because all solutions should be mathematically equivalent, some
|
||
|
|
hints may return the exact same result for an ODE. Often, though,
|
||
|
|
two different hints will return the same solution formatted
|
||
|
|
differently. The two should be equivalent. Also note that sometimes
|
||
|
|
the values of the arbitrary constants in two different solutions may
|
||
|
|
not be the same, because one constant may have "absorbed" other
|
||
|
|
constants into it.
|
||
|
|
- Do ``help(ode.ode_<hintname>)`` to get help more information on a
|
||
|
|
specific hint, where ``<hintname>`` is the name of a hint without
|
||
|
|
``_Integral``.
|
||
|
|
|
||
|
|
For system of ordinary differential equations
|
||
|
|
=============================================
|
||
|
|
|
||
|
|
**Usage**
|
||
|
|
``dsolve(eq, func)`` -> Solve a system of ordinary differential
|
||
|
|
equations ``eq`` for ``func`` being list of functions including
|
||
|
|
`x(t)`, `y(t)`, `z(t)` where number of functions in the list depends
|
||
|
|
upon the number of equations provided in ``eq``.
|
||
|
|
|
||
|
|
**Details**
|
||
|
|
|
||
|
|
``eq`` can be any supported system of ordinary differential equations
|
||
|
|
This can either be an :py:class:`~sympy.core.relational.Equality`,
|
||
|
|
or an expression, which is assumed to be equal to ``0``.
|
||
|
|
|
||
|
|
``func`` holds ``x(t)`` and ``y(t)`` being functions of one variable which
|
||
|
|
together with some of their derivatives make up the system of ordinary
|
||
|
|
differential equation ``eq``. It is not necessary to provide this; it
|
||
|
|
will be autodetected (and an error raised if it could not be detected).
|
||
|
|
|
||
|
|
**Hints**
|
||
|
|
|
||
|
|
The hints are formed by parameters returned by classify_sysode, combining
|
||
|
|
them give hints name used later for forming method name.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
>>> from sympy import Function, dsolve, Eq, Derivative, sin, cos, symbols
|
||
|
|
>>> from sympy.abc import x
|
||
|
|
>>> f = Function('f')
|
||
|
|
>>> dsolve(Derivative(f(x), x, x) + 9*f(x), f(x))
|
||
|
|
Eq(f(x), C1*sin(3*x) + C2*cos(3*x))
|
||
|
|
|
||
|
|
>>> eq = sin(x)*cos(f(x)) + cos(x)*sin(f(x))*f(x).diff(x)
|
||
|
|
>>> dsolve(eq, hint='1st_exact')
|
||
|
|
[Eq(f(x), -acos(C1/cos(x)) + 2*pi), Eq(f(x), acos(C1/cos(x)))]
|
||
|
|
>>> dsolve(eq, hint='almost_linear')
|
||
|
|
[Eq(f(x), -acos(C1/cos(x)) + 2*pi), Eq(f(x), acos(C1/cos(x)))]
|
||
|
|
>>> t = symbols('t')
|
||
|
|
>>> x, y = symbols('x, y', cls=Function)
|
||
|
|
>>> eq = (Eq(Derivative(x(t),t), 12*t*x(t) + 8*y(t)), Eq(Derivative(y(t),t), 21*x(t) + 7*t*y(t)))
|
||
|
|
>>> dsolve(eq)
|
||
|
|
[Eq(x(t), C1*x0(t) + C2*x0(t)*Integral(8*exp(Integral(7*t, t))*exp(Integral(12*t, t))/x0(t)**2, t)),
|
||
|
|
Eq(y(t), C1*y0(t) + C2*(y0(t)*Integral(8*exp(Integral(7*t, t))*exp(Integral(12*t, t))/x0(t)**2, t) +
|
||
|
|
exp(Integral(7*t, t))*exp(Integral(12*t, t))/x0(t)))]
|
||
|
|
>>> eq = (Eq(Derivative(x(t),t),x(t)*y(t)*sin(t)), Eq(Derivative(y(t),t),y(t)**2*sin(t)))
|
||
|
|
>>> dsolve(eq)
|
||
|
|
{Eq(x(t), -exp(C1)/(C2*exp(C1) - cos(t))), Eq(y(t), -1/(C1 - cos(t)))}
|
||
|
|
"""
|
||
|
|
if iterable(eq):
|
||
|
|
from sympy.solvers.ode.systems import dsolve_system
|
||
|
|
|
||
|
|
# This may have to be changed in future
|
||
|
|
# when we have weakly and strongly
|
||
|
|
# connected components. This have to
|
||
|
|
# changed to show the systems that haven't
|
||
|
|
# been solved.
|
||
|
|
try:
|
||
|
|
sol = dsolve_system(eq, funcs=func, ics=ics, doit=True)
|
||
|
|
return sol[0] if len(sol) == 1 else sol
|
||
|
|
except NotImplementedError:
|
||
|
|
pass
|
||
|
|
|
||
|
|
match = classify_sysode(eq, func)
|
||
|
|
|
||
|
|
eq = match['eq']
|
||
|
|
order = match['order']
|
||
|
|
func = match['func']
|
||
|
|
t = list(list(eq[0].atoms(Derivative))[0].atoms(Symbol))[0]
|
||
|
|
|
||
|
|
# keep highest order term coefficient positive
|
||
|
|
for i in range(len(eq)):
|
||
|
|
for func_ in func:
|
||
|
|
if isinstance(func_, list):
|
||
|
|
pass
|
||
|
|
else:
|
||
|
|
if eq[i].coeff(diff(func[i],t,ode_order(eq[i], func[i]))).is_negative:
|
||
|
|
eq[i] = -eq[i]
|
||
|
|
match['eq'] = eq
|
||
|
|
if len(set(order.values()))!=1:
|
||
|
|
raise ValueError("It solves only those systems of equations whose orders are equal")
|
||
|
|
match['order'] = list(order.values())[0]
|
||
|
|
def recur_len(l):
|
||
|
|
return sum(recur_len(item) if isinstance(item,list) else 1 for item in l)
|
||
|
|
if recur_len(func) != len(eq):
|
||
|
|
raise ValueError("dsolve() and classify_sysode() work with "
|
||
|
|
"number of functions being equal to number of equations")
|
||
|
|
if match['type_of_equation'] is None:
|
||
|
|
raise NotImplementedError
|
||
|
|
else:
|
||
|
|
if match['is_linear'] == True:
|
||
|
|
solvefunc = globals()['sysode_linear_%(no_of_equation)seq_order%(order)s' % match]
|
||
|
|
else:
|
||
|
|
solvefunc = globals()['sysode_nonlinear_%(no_of_equation)seq_order%(order)s' % match]
|
||
|
|
sols = solvefunc(match)
|
||
|
|
if ics:
|
||
|
|
constants = Tuple(*sols).free_symbols - Tuple(*eq).free_symbols
|
||
|
|
solved_constants = solve_ics(sols, func, constants, ics)
|
||
|
|
return [sol.subs(solved_constants) for sol in sols]
|
||
|
|
return sols
|
||
|
|
else:
|
||
|
|
given_hint = hint # hint given by the user
|
||
|
|
|
||
|
|
# See the docstring of _desolve for more details.
|
||
|
|
hints = _desolve(eq, func=func,
|
||
|
|
hint=hint, simplify=True, xi=xi, eta=eta, type='ode', ics=ics,
|
||
|
|
x0=x0, n=n, **kwargs)
|
||
|
|
eq = hints.pop('eq', eq)
|
||
|
|
all_ = hints.pop('all', False)
|
||
|
|
if all_:
|
||
|
|
retdict = {}
|
||
|
|
failed_hints = {}
|
||
|
|
gethints = classify_ode(eq, dict=True, hint='all')
|
||
|
|
orderedhints = gethints['ordered_hints']
|
||
|
|
for hint in hints:
|
||
|
|
try:
|
||
|
|
rv = _helper_simplify(eq, hint, hints[hint], simplify)
|
||
|
|
except NotImplementedError as detail:
|
||
|
|
failed_hints[hint] = detail
|
||
|
|
else:
|
||
|
|
retdict[hint] = rv
|
||
|
|
func = hints[hint]['func']
|
||
|
|
|
||
|
|
retdict['best'] = min(list(retdict.values()), key=lambda x:
|
||
|
|
ode_sol_simplicity(x, func, trysolving=not simplify))
|
||
|
|
if given_hint == 'best':
|
||
|
|
return retdict['best']
|
||
|
|
for i in orderedhints:
|
||
|
|
if retdict['best'] == retdict.get(i, None):
|
||
|
|
retdict['best_hint'] = i
|
||
|
|
break
|
||
|
|
retdict['default'] = gethints['default']
|
||
|
|
retdict['order'] = gethints['order']
|
||
|
|
retdict.update(failed_hints)
|
||
|
|
return retdict
|
||
|
|
|
||
|
|
else:
|
||
|
|
# The key 'hint' stores the hint needed to be solved for.
|
||
|
|
hint = hints['hint']
|
||
|
|
return _helper_simplify(eq, hint, hints, simplify, ics=ics)
|
||
|
|
|
||
|
|
def _helper_simplify(eq, hint, match, simplify=True, ics=None, **kwargs):
|
||
|
|
r"""
|
||
|
|
Helper function of dsolve that calls the respective
|
||
|
|
:py:mod:`~sympy.solvers.ode` functions to solve for the ordinary
|
||
|
|
differential equations. This minimizes the computation in calling
|
||
|
|
:py:meth:`~sympy.solvers.deutils._desolve` multiple times.
|
||
|
|
"""
|
||
|
|
r = match
|
||
|
|
func = r['func']
|
||
|
|
order = r['order']
|
||
|
|
match = r[hint]
|
||
|
|
|
||
|
|
if isinstance(match, SingleODESolver):
|
||
|
|
solvefunc = match
|
||
|
|
elif hint.endswith('_Integral'):
|
||
|
|
solvefunc = globals()['ode_' + hint[:-len('_Integral')]]
|
||
|
|
else:
|
||
|
|
solvefunc = globals()['ode_' + hint]
|
||
|
|
|
||
|
|
free = eq.free_symbols
|
||
|
|
cons = lambda s: s.free_symbols.difference(free)
|
||
|
|
|
||
|
|
if simplify:
|
||
|
|
# odesimp() will attempt to integrate, if necessary, apply constantsimp(),
|
||
|
|
# attempt to solve for func, and apply any other hint specific
|
||
|
|
# simplifications
|
||
|
|
if isinstance(solvefunc, SingleODESolver):
|
||
|
|
sols = solvefunc.get_general_solution()
|
||
|
|
else:
|
||
|
|
sols = solvefunc(eq, func, order, match)
|
||
|
|
if iterable(sols):
|
||
|
|
rv = [odesimp(eq, s, func, hint) for s in sols]
|
||
|
|
else:
|
||
|
|
rv = odesimp(eq, sols, func, hint)
|
||
|
|
else:
|
||
|
|
# We still want to integrate (you can disable it separately with the hint)
|
||
|
|
if isinstance(solvefunc, SingleODESolver):
|
||
|
|
exprs = solvefunc.get_general_solution(simplify=False)
|
||
|
|
else:
|
||
|
|
match['simplify'] = False # Some hints can take advantage of this option
|
||
|
|
exprs = solvefunc(eq, func, order, match)
|
||
|
|
if isinstance(exprs, list):
|
||
|
|
rv = [_handle_Integral(expr, func, hint) for expr in exprs]
|
||
|
|
else:
|
||
|
|
rv = _handle_Integral(exprs, func, hint)
|
||
|
|
|
||
|
|
if isinstance(rv, list):
|
||
|
|
if simplify:
|
||
|
|
rv = _remove_redundant_solutions(eq, rv, order, func.args[0])
|
||
|
|
if len(rv) == 1:
|
||
|
|
rv = rv[0]
|
||
|
|
if ics and 'power_series' not in hint:
|
||
|
|
if isinstance(rv, (Expr, Eq)):
|
||
|
|
solved_constants = solve_ics([rv], [r['func']], cons(rv), ics)
|
||
|
|
rv = rv.subs(solved_constants)
|
||
|
|
else:
|
||
|
|
rv1 = []
|
||
|
|
for s in rv:
|
||
|
|
try:
|
||
|
|
solved_constants = solve_ics([s], [r['func']], cons(s), ics)
|
||
|
|
except ValueError:
|
||
|
|
continue
|
||
|
|
rv1.append(s.subs(solved_constants))
|
||
|
|
if len(rv1) == 1:
|
||
|
|
return rv1[0]
|
||
|
|
rv = rv1
|
||
|
|
return rv
|
||
|
|
|
||
|
|
def solve_ics(sols, funcs, constants, ics):
|
||
|
|
"""
|
||
|
|
Solve for the constants given initial conditions
|
||
|
|
|
||
|
|
``sols`` is a list of solutions.
|
||
|
|
|
||
|
|
``funcs`` is a list of functions.
|
||
|
|
|
||
|
|
``constants`` is a list of constants.
|
||
|
|
|
||
|
|
``ics`` is the set of initial/boundary conditions for the differential
|
||
|
|
equation. It should be given in the form of ``{f(x0): x1,
|
||
|
|
f(x).diff(x).subs(x, x2): x3}`` and so on.
|
||
|
|
|
||
|
|
Returns a dictionary mapping constants to values.
|
||
|
|
``solution.subs(constants)`` will replace the constants in ``solution``.
|
||
|
|
|
||
|
|
Example
|
||
|
|
=======
|
||
|
|
>>> # From dsolve(f(x).diff(x) - f(x), f(x))
|
||
|
|
>>> from sympy import symbols, Eq, exp, Function
|
||
|
|
>>> from sympy.solvers.ode.ode import solve_ics
|
||
|
|
>>> f = Function('f')
|
||
|
|
>>> x, C1 = symbols('x C1')
|
||
|
|
>>> sols = [Eq(f(x), C1*exp(x))]
|
||
|
|
>>> funcs = [f(x)]
|
||
|
|
>>> constants = [C1]
|
||
|
|
>>> ics = {f(0): 2}
|
||
|
|
>>> solved_constants = solve_ics(sols, funcs, constants, ics)
|
||
|
|
>>> solved_constants
|
||
|
|
{C1: 2}
|
||
|
|
>>> sols[0].subs(solved_constants)
|
||
|
|
Eq(f(x), 2*exp(x))
|
||
|
|
|
||
|
|
"""
|
||
|
|
# Assume ics are of the form f(x0): value or Subs(diff(f(x), x, n), (x,
|
||
|
|
# x0)): value (currently checked by classify_ode). To solve, replace x
|
||
|
|
# with x0, f(x0) with value, then solve for constants. For f^(n)(x0),
|
||
|
|
# differentiate the solution n times, so that f^(n)(x) appears.
|
||
|
|
x = funcs[0].args[0]
|
||
|
|
diff_sols = []
|
||
|
|
subs_sols = []
|
||
|
|
diff_variables = set()
|
||
|
|
for funcarg, value in ics.items():
|
||
|
|
if isinstance(funcarg, AppliedUndef):
|
||
|
|
x0 = funcarg.args[0]
|
||
|
|
matching_func = [f for f in funcs if f.func == funcarg.func][0]
|
||
|
|
S = sols
|
||
|
|
elif isinstance(funcarg, (Subs, Derivative)):
|
||
|
|
if isinstance(funcarg, Subs):
|
||
|
|
# Make sure it stays a subs. Otherwise subs below will produce
|
||
|
|
# a different looking term.
|
||
|
|
funcarg = funcarg.doit()
|
||
|
|
if isinstance(funcarg, Subs):
|
||
|
|
deriv = funcarg.expr
|
||
|
|
x0 = funcarg.point[0]
|
||
|
|
variables = funcarg.expr.variables
|
||
|
|
matching_func = deriv
|
||
|
|
elif isinstance(funcarg, Derivative):
|
||
|
|
deriv = funcarg
|
||
|
|
x0 = funcarg.variables[0]
|
||
|
|
variables = (x,)*len(funcarg.variables)
|
||
|
|
matching_func = deriv.subs(x0, x)
|
||
|
|
for sol in sols:
|
||
|
|
if sol.has(deriv.expr.func):
|
||
|
|
diff_sols.append(Eq(sol.lhs.diff(*variables), sol.rhs.diff(*variables)))
|
||
|
|
diff_variables.add(variables)
|
||
|
|
S = diff_sols
|
||
|
|
else:
|
||
|
|
raise NotImplementedError("Unrecognized initial condition")
|
||
|
|
|
||
|
|
for sol in S:
|
||
|
|
if sol.has(matching_func):
|
||
|
|
sol2 = sol
|
||
|
|
sol2 = sol2.subs(x, x0)
|
||
|
|
sol2 = sol2.subs(funcarg, value)
|
||
|
|
# This check is necessary because of issue #15724
|
||
|
|
if not isinstance(sol2, BooleanAtom) or not subs_sols:
|
||
|
|
subs_sols = [s for s in subs_sols if not isinstance(s, BooleanAtom)]
|
||
|
|
subs_sols.append(sol2)
|
||
|
|
|
||
|
|
# TODO: Use solveset here
|
||
|
|
try:
|
||
|
|
solved_constants = solve(subs_sols, constants, dict=True)
|
||
|
|
except NotImplementedError:
|
||
|
|
solved_constants = []
|
||
|
|
|
||
|
|
# XXX: We can't differentiate between the solution not existing because of
|
||
|
|
# invalid initial conditions, and not existing because solve is not smart
|
||
|
|
# enough. If we could use solveset, this might be improvable, but for now,
|
||
|
|
# we use NotImplementedError in this case.
|
||
|
|
if not solved_constants:
|
||
|
|
raise ValueError("Couldn't solve for initial conditions")
|
||
|
|
|
||
|
|
if solved_constants == True:
|
||
|
|
raise ValueError("Initial conditions did not produce any solutions for constants. Perhaps they are degenerate.")
|
||
|
|
|
||
|
|
if len(solved_constants) > 1:
|
||
|
|
raise NotImplementedError("Initial conditions produced too many solutions for constants")
|
||
|
|
|
||
|
|
return solved_constants[0]
|
||
|
|
|
||
|
|
def classify_ode(eq, func=None, dict=False, ics=None, *, prep=True, xi=None, eta=None, n=None, **kwargs):
|
||
|
|
r"""
|
||
|
|
Returns a tuple of possible :py:meth:`~sympy.solvers.ode.dsolve`
|
||
|
|
classifications for an ODE.
|
||
|
|
|
||
|
|
The tuple is ordered so that first item is the classification that
|
||
|
|
:py:meth:`~sympy.solvers.ode.dsolve` uses to solve the ODE by default. In
|
||
|
|
general, classifications at the near the beginning of the list will
|
||
|
|
produce better solutions faster than those near the end, thought there are
|
||
|
|
always exceptions. To make :py:meth:`~sympy.solvers.ode.dsolve` use a
|
||
|
|
different classification, use ``dsolve(ODE, func,
|
||
|
|
hint=<classification>)``. See also the
|
||
|
|
:py:meth:`~sympy.solvers.ode.dsolve` docstring for different meta-hints
|
||
|
|
you can use.
|
||
|
|
|
||
|
|
If ``dict`` is true, :py:meth:`~sympy.solvers.ode.classify_ode` will
|
||
|
|
return a dictionary of ``hint:match`` expression terms. This is intended
|
||
|
|
for internal use by :py:meth:`~sympy.solvers.ode.dsolve`. Note that
|
||
|
|
because dictionaries are ordered arbitrarily, this will most likely not be
|
||
|
|
in the same order as the tuple.
|
||
|
|
|
||
|
|
You can get help on different hints by executing
|
||
|
|
``help(ode.ode_hintname)``, where ``hintname`` is the name of the hint
|
||
|
|
without ``_Integral``.
|
||
|
|
|
||
|
|
See :py:data:`~sympy.solvers.ode.allhints` or the
|
||
|
|
:py:mod:`~sympy.solvers.ode` docstring for a list of all supported hints
|
||
|
|
that can be returned from :py:meth:`~sympy.solvers.ode.classify_ode`.
|
||
|
|
|
||
|
|
Notes
|
||
|
|
=====
|
||
|
|
|
||
|
|
These are remarks on hint names.
|
||
|
|
|
||
|
|
``_Integral``
|
||
|
|
|
||
|
|
If a classification has ``_Integral`` at the end, it will return the
|
||
|
|
expression with an unevaluated :py:class:`~.Integral`
|
||
|
|
class in it. Note that a hint may do this anyway if
|
||
|
|
:py:meth:`~sympy.core.expr.Expr.integrate` cannot do the integral,
|
||
|
|
though just using an ``_Integral`` will do so much faster. Indeed, an
|
||
|
|
``_Integral`` hint will always be faster than its corresponding hint
|
||
|
|
without ``_Integral`` because
|
||
|
|
:py:meth:`~sympy.core.expr.Expr.integrate` is an expensive routine.
|
||
|
|
If :py:meth:`~sympy.solvers.ode.dsolve` hangs, it is probably because
|
||
|
|
:py:meth:`~sympy.core.expr.Expr.integrate` is hanging on a tough or
|
||
|
|
impossible integral. Try using an ``_Integral`` hint or
|
||
|
|
``all_Integral`` to get it return something.
|
||
|
|
|
||
|
|
Note that some hints do not have ``_Integral`` counterparts. This is
|
||
|
|
because :py:func:`~sympy.integrals.integrals.integrate` is not used in
|
||
|
|
solving the ODE for those method. For example, `n`\th order linear
|
||
|
|
homogeneous ODEs with constant coefficients do not require integration
|
||
|
|
to solve, so there is no
|
||
|
|
``nth_linear_homogeneous_constant_coeff_Integrate`` hint. You can
|
||
|
|
easily evaluate any unevaluated
|
||
|
|
:py:class:`~sympy.integrals.integrals.Integral`\s in an expression by
|
||
|
|
doing ``expr.doit()``.
|
||
|
|
|
||
|
|
Ordinals
|
||
|
|
|
||
|
|
Some hints contain an ordinal such as ``1st_linear``. This is to help
|
||
|
|
differentiate them from other hints, as well as from other methods
|
||
|
|
that may not be implemented yet. If a hint has ``nth`` in it, such as
|
||
|
|
the ``nth_linear`` hints, this means that the method used to applies
|
||
|
|
to ODEs of any order.
|
||
|
|
|
||
|
|
``indep`` and ``dep``
|
||
|
|
|
||
|
|
Some hints contain the words ``indep`` or ``dep``. These reference
|
||
|
|
the independent variable and the dependent function, respectively. For
|
||
|
|
example, if an ODE is in terms of `f(x)`, then ``indep`` will refer to
|
||
|
|
`x` and ``dep`` will refer to `f`.
|
||
|
|
|
||
|
|
``subs``
|
||
|
|
|
||
|
|
If a hints has the word ``subs`` in it, it means that the ODE is solved
|
||
|
|
by substituting the expression given after the word ``subs`` for a
|
||
|
|
single dummy variable. This is usually in terms of ``indep`` and
|
||
|
|
``dep`` as above. The substituted expression will be written only in
|
||
|
|
characters allowed for names of Python objects, meaning operators will
|
||
|
|
be spelled out. For example, ``indep``/``dep`` will be written as
|
||
|
|
``indep_div_dep``.
|
||
|
|
|
||
|
|
``coeff``
|
||
|
|
|
||
|
|
The word ``coeff`` in a hint refers to the coefficients of something
|
||
|
|
in the ODE, usually of the derivative terms. See the docstring for
|
||
|
|
the individual methods for more info (``help(ode)``). This is
|
||
|
|
contrast to ``coefficients``, as in ``undetermined_coefficients``,
|
||
|
|
which refers to the common name of a method.
|
||
|
|
|
||
|
|
``_best``
|
||
|
|
|
||
|
|
Methods that have more than one fundamental way to solve will have a
|
||
|
|
hint for each sub-method and a ``_best`` meta-classification. This
|
||
|
|
will evaluate all hints and return the best, using the same
|
||
|
|
considerations as the normal ``best`` meta-hint.
|
||
|
|
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
>>> from sympy import Function, classify_ode, Eq
|
||
|
|
>>> from sympy.abc import x
|
||
|
|
>>> f = Function('f')
|
||
|
|
>>> classify_ode(Eq(f(x).diff(x), 0), f(x))
|
||
|
|
('nth_algebraic',
|
||
|
|
'separable',
|
||
|
|
'1st_exact',
|
||
|
|
'1st_linear',
|
||
|
|
'Bernoulli',
|
||
|
|
'1st_homogeneous_coeff_best',
|
||
|
|
'1st_homogeneous_coeff_subs_indep_div_dep',
|
||
|
|
'1st_homogeneous_coeff_subs_dep_div_indep',
|
||
|
|
'1st_power_series', 'lie_group', 'nth_linear_constant_coeff_homogeneous',
|
||
|
|
'nth_linear_euler_eq_homogeneous',
|
||
|
|
'nth_algebraic_Integral', 'separable_Integral', '1st_exact_Integral',
|
||
|
|
'1st_linear_Integral', 'Bernoulli_Integral',
|
||
|
|
'1st_homogeneous_coeff_subs_indep_div_dep_Integral',
|
||
|
|
'1st_homogeneous_coeff_subs_dep_div_indep_Integral')
|
||
|
|
>>> classify_ode(f(x).diff(x, 2) + 3*f(x).diff(x) + 2*f(x) - 4)
|
||
|
|
('factorable', 'nth_linear_constant_coeff_undetermined_coefficients',
|
||
|
|
'nth_linear_constant_coeff_variation_of_parameters',
|
||
|
|
'nth_linear_constant_coeff_variation_of_parameters_Integral')
|
||
|
|
|
||
|
|
"""
|
||
|
|
ics = sympify(ics)
|
||
|
|
|
||
|
|
if func and len(func.args) != 1:
|
||
|
|
raise ValueError("dsolve() and classify_ode() only "
|
||
|
|
"work with functions of one variable, not %s" % func)
|
||
|
|
|
||
|
|
if isinstance(eq, Equality):
|
||
|
|
eq = eq.lhs - eq.rhs
|
||
|
|
|
||
|
|
# Some methods want the unprocessed equation
|
||
|
|
eq_orig = eq
|
||
|
|
|
||
|
|
if prep or func is None:
|
||
|
|
eq, func_ = _preprocess(eq, func)
|
||
|
|
if func is None:
|
||
|
|
func = func_
|
||
|
|
x = func.args[0]
|
||
|
|
f = func.func
|
||
|
|
y = Dummy('y')
|
||
|
|
terms = 5 if n is None else n
|
||
|
|
|
||
|
|
order = ode_order(eq, f(x))
|
||
|
|
# hint:matchdict or hint:(tuple of matchdicts)
|
||
|
|
# Also will contain "default":<default hint> and "order":order items.
|
||
|
|
matching_hints = {"order": order}
|
||
|
|
|
||
|
|
df = f(x).diff(x)
|
||
|
|
a = Wild('a', exclude=[f(x)])
|
||
|
|
d = Wild('d', exclude=[df, f(x).diff(x, 2)])
|
||
|
|
e = Wild('e', exclude=[df])
|
||
|
|
n = Wild('n', exclude=[x, f(x), df])
|
||
|
|
c1 = Wild('c1', exclude=[x])
|
||
|
|
a3 = Wild('a3', exclude=[f(x), df, f(x).diff(x, 2)])
|
||
|
|
b3 = Wild('b3', exclude=[f(x), df, f(x).diff(x, 2)])
|
||
|
|
c3 = Wild('c3', exclude=[f(x), df, f(x).diff(x, 2)])
|
||
|
|
boundary = {} # Used to extract initial conditions
|
||
|
|
C1 = Symbol("C1")
|
||
|
|
|
||
|
|
# Preprocessing to get the initial conditions out
|
||
|
|
if ics is not None:
|
||
|
|
for funcarg in ics:
|
||
|
|
# Separating derivatives
|
||
|
|
if isinstance(funcarg, (Subs, Derivative)):
|
||
|
|
# f(x).diff(x).subs(x, 0) is a Subs, but f(x).diff(x).subs(x,
|
||
|
|
# y) is a Derivative
|
||
|
|
if isinstance(funcarg, Subs):
|
||
|
|
deriv = funcarg.expr
|
||
|
|
old = funcarg.variables[0]
|
||
|
|
new = funcarg.point[0]
|
||
|
|
elif isinstance(funcarg, Derivative):
|
||
|
|
deriv = funcarg
|
||
|
|
# No information on this. Just assume it was x
|
||
|
|
old = x
|
||
|
|
new = funcarg.variables[0]
|
||
|
|
|
||
|
|
if (isinstance(deriv, Derivative) and isinstance(deriv.args[0],
|
||
|
|
AppliedUndef) and deriv.args[0].func == f and
|
||
|
|
len(deriv.args[0].args) == 1 and old == x and not
|
||
|
|
new.has(x) and all(i == deriv.variables[0] for i in
|
||
|
|
deriv.variables) and x not in ics[funcarg].free_symbols):
|
||
|
|
|
||
|
|
dorder = ode_order(deriv, x)
|
||
|
|
temp = 'f' + str(dorder)
|
||
|
|
boundary.update({temp: new, temp + 'val': ics[funcarg]})
|
||
|
|
else:
|
||
|
|
raise ValueError("Invalid boundary conditions for Derivatives")
|
||
|
|
|
||
|
|
|
||
|
|
# Separating functions
|
||
|
|
elif isinstance(funcarg, AppliedUndef):
|
||
|
|
if (funcarg.func == f and len(funcarg.args) == 1 and
|
||
|
|
not funcarg.args[0].has(x) and x not in ics[funcarg].free_symbols):
|
||
|
|
boundary.update({'f0': funcarg.args[0], 'f0val': ics[funcarg]})
|
||
|
|
else:
|
||
|
|
raise ValueError("Invalid boundary conditions for Function")
|
||
|
|
|
||
|
|
else:
|
||
|
|
raise ValueError("Enter boundary conditions of the form ics={f(point): value, f(x).diff(x, order).subs(x, point): value}")
|
||
|
|
|
||
|
|
ode = SingleODEProblem(eq_orig, func, x, prep=prep, xi=xi, eta=eta)
|
||
|
|
user_hint = kwargs.get('hint', 'default')
|
||
|
|
# Used when dsolve is called without an explicit hint.
|
||
|
|
# We exit early to return the first valid match
|
||
|
|
early_exit = (user_hint=='default')
|
||
|
|
if user_hint.endswith('_Integral'):
|
||
|
|
user_hint = user_hint[:-len('_Integral')]
|
||
|
|
user_map = solver_map
|
||
|
|
# An explicit hint has been given to dsolve
|
||
|
|
# Skip matching code for other hints
|
||
|
|
if user_hint not in ['default', 'all', 'all_Integral', 'best'] and user_hint in solver_map:
|
||
|
|
user_map = {user_hint: solver_map[user_hint]}
|
||
|
|
|
||
|
|
for hint in user_map:
|
||
|
|
solver = user_map[hint](ode)
|
||
|
|
if solver.matches():
|
||
|
|
matching_hints[hint] = solver
|
||
|
|
if user_map[hint].has_integral:
|
||
|
|
matching_hints[hint + "_Integral"] = solver
|
||
|
|
if dict and early_exit:
|
||
|
|
matching_hints["default"] = hint
|
||
|
|
return matching_hints
|
||
|
|
|
||
|
|
eq = expand(eq)
|
||
|
|
# Precondition to try remove f(x) from highest order derivative
|
||
|
|
reduced_eq = None
|
||
|
|
if eq.is_Add:
|
||
|
|
deriv_coef = eq.coeff(f(x).diff(x, order))
|
||
|
|
if deriv_coef not in (1, 0):
|
||
|
|
r = deriv_coef.match(a*f(x)**c1)
|
||
|
|
if r and r[c1]:
|
||
|
|
den = f(x)**r[c1]
|
||
|
|
reduced_eq = Add(*[arg/den for arg in eq.args])
|
||
|
|
if not reduced_eq:
|
||
|
|
reduced_eq = eq
|
||
|
|
|
||
|
|
if order == 1:
|
||
|
|
|
||
|
|
# NON-REDUCED FORM OF EQUATION matches
|
||
|
|
r = collect(eq, df, exact=True).match(d + e * df)
|
||
|
|
if r:
|
||
|
|
r['d'] = d
|
||
|
|
r['e'] = e
|
||
|
|
r['y'] = y
|
||
|
|
r[d] = r[d].subs(f(x), y)
|
||
|
|
r[e] = r[e].subs(f(x), y)
|
||
|
|
|
||
|
|
# FIRST ORDER POWER SERIES WHICH NEEDS INITIAL CONDITIONS
|
||
|
|
# TODO: Hint first order series should match only if d/e is analytic.
|
||
|
|
# For now, only d/e and (d/e).diff(arg) is checked for existence at
|
||
|
|
# at a given point.
|
||
|
|
# This is currently done internally in ode_1st_power_series.
|
||
|
|
point = boundary.get('f0', 0)
|
||
|
|
value = boundary.get('f0val', C1)
|
||
|
|
check = cancel(r[d]/r[e])
|
||
|
|
check1 = check.subs({x: point, y: value})
|
||
|
|
if not check1.has(oo) and not check1.has(zoo) and \
|
||
|
|
not check1.has(nan) and not check1.has(-oo):
|
||
|
|
check2 = (check1.diff(x)).subs({x: point, y: value})
|
||
|
|
if not check2.has(oo) and not check2.has(zoo) and \
|
||
|
|
not check2.has(nan) and not check2.has(-oo):
|
||
|
|
rseries = r.copy()
|
||
|
|
rseries.update({'terms': terms, 'f0': point, 'f0val': value})
|
||
|
|
matching_hints["1st_power_series"] = rseries
|
||
|
|
|
||
|
|
elif order == 2:
|
||
|
|
# Homogeneous second order differential equation of the form
|
||
|
|
# a3*f(x).diff(x, 2) + b3*f(x).diff(x) + c3
|
||
|
|
# It has a definite power series solution at point x0 if, b3/a3 and c3/a3
|
||
|
|
# are analytic at x0.
|
||
|
|
deq = a3*(f(x).diff(x, 2)) + b3*df + c3*f(x)
|
||
|
|
r = collect(reduced_eq,
|
||
|
|
[f(x).diff(x, 2), f(x).diff(x), f(x)]).match(deq)
|
||
|
|
ordinary = False
|
||
|
|
if r:
|
||
|
|
if not all(r[key].is_polynomial() for key in r):
|
||
|
|
n, d = reduced_eq.as_numer_denom()
|
||
|
|
reduced_eq = expand(n)
|
||
|
|
r = collect(reduced_eq,
|
||
|
|
[f(x).diff(x, 2), f(x).diff(x), f(x)]).match(deq)
|
||
|
|
if r and r[a3] != 0:
|
||
|
|
p = cancel(r[b3]/r[a3]) # Used below
|
||
|
|
q = cancel(r[c3]/r[a3]) # Used below
|
||
|
|
point = kwargs.get('x0', 0)
|
||
|
|
check = p.subs(x, point)
|
||
|
|
if not check.has(oo, nan, zoo, -oo):
|
||
|
|
check = q.subs(x, point)
|
||
|
|
if not check.has(oo, nan, zoo, -oo):
|
||
|
|
ordinary = True
|
||
|
|
r.update({'a3': a3, 'b3': b3, 'c3': c3, 'x0': point, 'terms': terms})
|
||
|
|
matching_hints["2nd_power_series_ordinary"] = r
|
||
|
|
|
||
|
|
# Checking if the differential equation has a regular singular point
|
||
|
|
# at x0. It has a regular singular point at x0, if (b3/a3)*(x - x0)
|
||
|
|
# and (c3/a3)*((x - x0)**2) are analytic at x0.
|
||
|
|
if not ordinary:
|
||
|
|
p = cancel((x - point)*p)
|
||
|
|
check = p.subs(x, point)
|
||
|
|
if not check.has(oo, nan, zoo, -oo):
|
||
|
|
q = cancel(((x - point)**2)*q)
|
||
|
|
check = q.subs(x, point)
|
||
|
|
if not check.has(oo, nan, zoo, -oo):
|
||
|
|
coeff_dict = {'p': p, 'q': q, 'x0': point, 'terms': terms}
|
||
|
|
matching_hints["2nd_power_series_regular"] = coeff_dict
|
||
|
|
|
||
|
|
|
||
|
|
# Order keys based on allhints.
|
||
|
|
retlist = [i for i in allhints if i in matching_hints]
|
||
|
|
if dict:
|
||
|
|
# Dictionaries are ordered arbitrarily, so make note of which
|
||
|
|
# hint would come first for dsolve(). Use an ordered dict in Py 3.
|
||
|
|
matching_hints["default"] = retlist[0] if retlist else None
|
||
|
|
matching_hints["ordered_hints"] = tuple(retlist)
|
||
|
|
return matching_hints
|
||
|
|
else:
|
||
|
|
return tuple(retlist)
|
||
|
|
|
||
|
|
|
||
|
|
def classify_sysode(eq, funcs=None, **kwargs):
|
||
|
|
r"""
|
||
|
|
Returns a dictionary of parameter names and values that define the system
|
||
|
|
of ordinary differential equations in ``eq``.
|
||
|
|
The parameters are further used in
|
||
|
|
:py:meth:`~sympy.solvers.ode.dsolve` for solving that system.
|
||
|
|
|
||
|
|
Some parameter names and values are:
|
||
|
|
|
||
|
|
'is_linear' (boolean), which tells whether the given system is linear.
|
||
|
|
Note that "linear" here refers to the operator: terms such as ``x*diff(x,t)`` are
|
||
|
|
nonlinear, whereas terms like ``sin(t)*diff(x,t)`` are still linear operators.
|
||
|
|
|
||
|
|
'func' (list) contains the :py:class:`~sympy.core.function.Function`s that
|
||
|
|
appear with a derivative in the ODE, i.e. those that we are trying to solve
|
||
|
|
the ODE for.
|
||
|
|
|
||
|
|
'order' (dict) with the maximum derivative for each element of the 'func'
|
||
|
|
parameter.
|
||
|
|
|
||
|
|
'func_coeff' (dict or Matrix) with the coefficient for each triple ``(equation number,
|
||
|
|
function, order)```. The coefficients are those subexpressions that do not
|
||
|
|
appear in 'func', and hence can be considered constant for purposes of ODE
|
||
|
|
solving. The value of this parameter can also be a Matrix if the system of ODEs are
|
||
|
|
linear first order of the form X' = AX where X is the vector of dependent variables.
|
||
|
|
Here, this function returns the coefficient matrix A.
|
||
|
|
|
||
|
|
'eq' (list) with the equations from ``eq``, sympified and transformed into
|
||
|
|
expressions (we are solving for these expressions to be zero).
|
||
|
|
|
||
|
|
'no_of_equations' (int) is the number of equations (same as ``len(eq)``).
|
||
|
|
|
||
|
|
'type_of_equation' (string) is an internal classification of the type of
|
||
|
|
ODE.
|
||
|
|
|
||
|
|
'is_constant' (boolean), which tells if the system of ODEs is constant coefficient
|
||
|
|
or not. This key is temporary addition for now and is in the match dict only when
|
||
|
|
the system of ODEs is linear first order constant coefficient homogeneous. So, this
|
||
|
|
key's value is True for now if it is available else it does not exist.
|
||
|
|
|
||
|
|
'is_homogeneous' (boolean), which tells if the system of ODEs is homogeneous. Like the
|
||
|
|
key 'is_constant', this key is a temporary addition and it is True since this key value
|
||
|
|
is available only when the system is linear first order constant coefficient homogeneous.
|
||
|
|
|
||
|
|
References
|
||
|
|
==========
|
||
|
|
-https://eqworld.ipmnet.ru/en/solutions/sysode/sode-toc1.htm
|
||
|
|
-A. D. Polyanin and A. V. Manzhirov, Handbook of Mathematics for Engineers and Scientists
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
>>> from sympy import Function, Eq, symbols, diff
|
||
|
|
>>> from sympy.solvers.ode.ode import classify_sysode
|
||
|
|
>>> from sympy.abc import t
|
||
|
|
>>> f, x, y = symbols('f, x, y', cls=Function)
|
||
|
|
>>> k, l, m, n = symbols('k, l, m, n', Integer=True)
|
||
|
|
>>> x1 = diff(x(t), t) ; y1 = diff(y(t), t)
|
||
|
|
>>> x2 = diff(x(t), t, t) ; y2 = diff(y(t), t, t)
|
||
|
|
>>> eq = (Eq(x1, 12*x(t) - 6*y(t)), Eq(y1, 11*x(t) + 3*y(t)))
|
||
|
|
>>> classify_sysode(eq)
|
||
|
|
{'eq': [-12*x(t) + 6*y(t) + Derivative(x(t), t), -11*x(t) - 3*y(t) + Derivative(y(t), t)], 'func': [x(t), y(t)],
|
||
|
|
'func_coeff': {(0, x(t), 0): -12, (0, x(t), 1): 1, (0, y(t), 0): 6, (0, y(t), 1): 0, (1, x(t), 0): -11, (1, x(t), 1): 0, (1, y(t), 0): -3, (1, y(t), 1): 1}, 'is_linear': True, 'no_of_equation': 2, 'order': {x(t): 1, y(t): 1}, 'type_of_equation': None}
|
||
|
|
>>> eq = (Eq(diff(x(t),t), 5*t*x(t) + t**2*y(t) + 2), Eq(diff(y(t),t), -t**2*x(t) + 5*t*y(t)))
|
||
|
|
>>> classify_sysode(eq)
|
||
|
|
{'eq': [-t**2*y(t) - 5*t*x(t) + Derivative(x(t), t) - 2, t**2*x(t) - 5*t*y(t) + Derivative(y(t), t)],
|
||
|
|
'func': [x(t), y(t)], 'func_coeff': {(0, x(t), 0): -5*t, (0, x(t), 1): 1, (0, y(t), 0): -t**2, (0, y(t), 1): 0,
|
||
|
|
(1, x(t), 0): t**2, (1, x(t), 1): 0, (1, y(t), 0): -5*t, (1, y(t), 1): 1}, 'is_linear': True, 'no_of_equation': 2,
|
||
|
|
'order': {x(t): 1, y(t): 1}, 'type_of_equation': None}
|
||
|
|
|
||
|
|
"""
|
||
|
|
|
||
|
|
# Sympify equations and convert iterables of equations into
|
||
|
|
# a list of equations
|
||
|
|
def _sympify(eq):
|
||
|
|
return list(map(sympify, eq if iterable(eq) else [eq]))
|
||
|
|
|
||
|
|
eq, funcs = (_sympify(w) for w in [eq, funcs])
|
||
|
|
for i, fi in enumerate(eq):
|
||
|
|
if isinstance(fi, Equality):
|
||
|
|
eq[i] = fi.lhs - fi.rhs
|
||
|
|
|
||
|
|
t = list(list(eq[0].atoms(Derivative))[0].atoms(Symbol))[0]
|
||
|
|
matching_hints = {"no_of_equation":i+1}
|
||
|
|
matching_hints['eq'] = eq
|
||
|
|
if i==0:
|
||
|
|
raise ValueError("classify_sysode() works for systems of ODEs. "
|
||
|
|
"For scalar ODEs, classify_ode should be used")
|
||
|
|
|
||
|
|
# find all the functions if not given
|
||
|
|
order = {}
|
||
|
|
if funcs==[None]:
|
||
|
|
funcs = _extract_funcs(eq)
|
||
|
|
|
||
|
|
funcs = list(set(funcs))
|
||
|
|
if len(funcs) != len(eq):
|
||
|
|
raise ValueError("Number of functions given is not equal to the number of equations %s" % funcs)
|
||
|
|
|
||
|
|
# This logic of list of lists in funcs to
|
||
|
|
# be replaced later.
|
||
|
|
func_dict = {}
|
||
|
|
for func in funcs:
|
||
|
|
if not order.get(func, False):
|
||
|
|
max_order = 0
|
||
|
|
for i, eqs_ in enumerate(eq):
|
||
|
|
order_ = ode_order(eqs_,func)
|
||
|
|
if max_order < order_:
|
||
|
|
max_order = order_
|
||
|
|
eq_no = i
|
||
|
|
if eq_no in func_dict:
|
||
|
|
func_dict[eq_no] = [func_dict[eq_no], func]
|
||
|
|
else:
|
||
|
|
func_dict[eq_no] = func
|
||
|
|
order[func] = max_order
|
||
|
|
|
||
|
|
funcs = [func_dict[i] for i in range(len(func_dict))]
|
||
|
|
matching_hints['func'] = funcs
|
||
|
|
for func in funcs:
|
||
|
|
if isinstance(func, list):
|
||
|
|
for func_elem in func:
|
||
|
|
if len(func_elem.args) != 1:
|
||
|
|
raise ValueError("dsolve() and classify_sysode() work with "
|
||
|
|
"functions of one variable only, not %s" % func)
|
||
|
|
else:
|
||
|
|
if func and len(func.args) != 1:
|
||
|
|
raise ValueError("dsolve() and classify_sysode() work with "
|
||
|
|
"functions of one variable only, not %s" % func)
|
||
|
|
|
||
|
|
# find the order of all equation in system of odes
|
||
|
|
matching_hints["order"] = order
|
||
|
|
|
||
|
|
# find coefficients of terms f(t), diff(f(t),t) and higher derivatives
|
||
|
|
# and similarly for other functions g(t), diff(g(t),t) in all equations.
|
||
|
|
# Here j denotes the equation number, funcs[l] denotes the function about
|
||
|
|
# which we are talking about and k denotes the order of function funcs[l]
|
||
|
|
# whose coefficient we are calculating.
|
||
|
|
def linearity_check(eqs, j, func, is_linear_):
|
||
|
|
for k in range(order[func] + 1):
|
||
|
|
func_coef[j, func, k] = collect(eqs.expand(), [diff(func, t, k)]).coeff(diff(func, t, k))
|
||
|
|
if is_linear_ == True:
|
||
|
|
if func_coef[j, func, k] == 0:
|
||
|
|
if k == 0:
|
||
|
|
coef = eqs.as_independent(func, as_Add=True)[1]
|
||
|
|
for xr in range(1, ode_order(eqs,func) + 1):
|
||
|
|
coef -= eqs.as_independent(diff(func, t, xr), as_Add=True)[1]
|
||
|
|
if coef != 0:
|
||
|
|
is_linear_ = False
|
||
|
|
else:
|
||
|
|
if eqs.as_independent(diff(func, t, k), as_Add=True)[1]:
|
||
|
|
is_linear_ = False
|
||
|
|
else:
|
||
|
|
for func_ in funcs:
|
||
|
|
if isinstance(func_, list):
|
||
|
|
for elem_func_ in func_:
|
||
|
|
dep = func_coef[j, func, k].as_independent(elem_func_, as_Add=True)[1]
|
||
|
|
if dep != 0:
|
||
|
|
is_linear_ = False
|
||
|
|
else:
|
||
|
|
dep = func_coef[j, func, k].as_independent(func_, as_Add=True)[1]
|
||
|
|
if dep != 0:
|
||
|
|
is_linear_ = False
|
||
|
|
return is_linear_
|
||
|
|
|
||
|
|
func_coef = {}
|
||
|
|
is_linear = True
|
||
|
|
for j, eqs in enumerate(eq):
|
||
|
|
for func in funcs:
|
||
|
|
if isinstance(func, list):
|
||
|
|
for func_elem in func:
|
||
|
|
is_linear = linearity_check(eqs, j, func_elem, is_linear)
|
||
|
|
else:
|
||
|
|
is_linear = linearity_check(eqs, j, func, is_linear)
|
||
|
|
matching_hints['func_coeff'] = func_coef
|
||
|
|
matching_hints['is_linear'] = is_linear
|
||
|
|
|
||
|
|
|
||
|
|
if len(set(order.values())) == 1:
|
||
|
|
order_eq = list(matching_hints['order'].values())[0]
|
||
|
|
if matching_hints['is_linear'] == True:
|
||
|
|
if matching_hints['no_of_equation'] == 2:
|
||
|
|
if order_eq == 1:
|
||
|
|
type_of_equation = check_linear_2eq_order1(eq, funcs, func_coef)
|
||
|
|
else:
|
||
|
|
type_of_equation = None
|
||
|
|
# If the equation does not match up with any of the
|
||
|
|
# general case solvers in systems.py and the number
|
||
|
|
# of equations is greater than 2, then NotImplementedError
|
||
|
|
# should be raised.
|
||
|
|
else:
|
||
|
|
type_of_equation = None
|
||
|
|
|
||
|
|
else:
|
||
|
|
if matching_hints['no_of_equation'] == 2:
|
||
|
|
if order_eq == 1:
|
||
|
|
type_of_equation = check_nonlinear_2eq_order1(eq, funcs, func_coef)
|
||
|
|
else:
|
||
|
|
type_of_equation = None
|
||
|
|
elif matching_hints['no_of_equation'] == 3:
|
||
|
|
if order_eq == 1:
|
||
|
|
type_of_equation = check_nonlinear_3eq_order1(eq, funcs, func_coef)
|
||
|
|
else:
|
||
|
|
type_of_equation = None
|
||
|
|
else:
|
||
|
|
type_of_equation = None
|
||
|
|
else:
|
||
|
|
type_of_equation = None
|
||
|
|
|
||
|
|
matching_hints['type_of_equation'] = type_of_equation
|
||
|
|
|
||
|
|
return matching_hints
|
||
|
|
|
||
|
|
|
||
|
|
def check_linear_2eq_order1(eq, func, func_coef):
|
||
|
|
x = func[0].func
|
||
|
|
y = func[1].func
|
||
|
|
fc = func_coef
|
||
|
|
t = list(list(eq[0].atoms(Derivative))[0].atoms(Symbol))[0]
|
||
|
|
r = {}
|
||
|
|
# for equations Eq(a1*diff(x(t),t), b1*x(t) + c1*y(t) + d1)
|
||
|
|
# and Eq(a2*diff(y(t),t), b2*x(t) + c2*y(t) + d2)
|
||
|
|
r['a1'] = fc[0,x(t),1] ; r['a2'] = fc[1,y(t),1]
|
||
|
|
r['b1'] = -fc[0,x(t),0]/fc[0,x(t),1] ; r['b2'] = -fc[1,x(t),0]/fc[1,y(t),1]
|
||
|
|
r['c1'] = -fc[0,y(t),0]/fc[0,x(t),1] ; r['c2'] = -fc[1,y(t),0]/fc[1,y(t),1]
|
||
|
|
forcing = [S.Zero,S.Zero]
|
||
|
|
for i in range(2):
|
||
|
|
for j in Add.make_args(eq[i]):
|
||
|
|
if not j.has(x(t), y(t)):
|
||
|
|
forcing[i] += j
|
||
|
|
if not (forcing[0].has(t) or forcing[1].has(t)):
|
||
|
|
# We can handle homogeneous case and simple constant forcings
|
||
|
|
r['d1'] = forcing[0]
|
||
|
|
r['d2'] = forcing[1]
|
||
|
|
else:
|
||
|
|
# Issue #9244: nonhomogeneous linear systems are not supported
|
||
|
|
return None
|
||
|
|
|
||
|
|
# Conditions to check for type 6 whose equations are Eq(diff(x(t),t), f(t)*x(t) + g(t)*y(t)) and
|
||
|
|
# Eq(diff(y(t),t), a*[f(t) + a*h(t)]x(t) + a*[g(t) - h(t)]*y(t))
|
||
|
|
p = 0
|
||
|
|
q = 0
|
||
|
|
p1 = cancel(r['b2']/(cancel(r['b2']/r['c2']).as_numer_denom()[0]))
|
||
|
|
p2 = cancel(r['b1']/(cancel(r['b1']/r['c1']).as_numer_denom()[0]))
|
||
|
|
for n, i in enumerate([p1, p2]):
|
||
|
|
for j in Mul.make_args(collect_const(i)):
|
||
|
|
if not j.has(t):
|
||
|
|
q = j
|
||
|
|
if q and n==0:
|
||
|
|
if ((r['b2']/j - r['b1'])/(r['c1'] - r['c2']/j)) == j:
|
||
|
|
p = 1
|
||
|
|
elif q and n==1:
|
||
|
|
if ((r['b1']/j - r['b2'])/(r['c2'] - r['c1']/j)) == j:
|
||
|
|
p = 2
|
||
|
|
# End of condition for type 6
|
||
|
|
|
||
|
|
if r['d1']!=0 or r['d2']!=0:
|
||
|
|
return None
|
||
|
|
else:
|
||
|
|
if not any(r[k].has(t) for k in 'a1 a2 b1 b2 c1 c2'.split()):
|
||
|
|
return None
|
||
|
|
else:
|
||
|
|
r['b1'] = r['b1']/r['a1'] ; r['b2'] = r['b2']/r['a2']
|
||
|
|
r['c1'] = r['c1']/r['a1'] ; r['c2'] = r['c2']/r['a2']
|
||
|
|
if p:
|
||
|
|
return "type6"
|
||
|
|
else:
|
||
|
|
# Equations for type 7 are Eq(diff(x(t),t), f(t)*x(t) + g(t)*y(t)) and Eq(diff(y(t),t), h(t)*x(t) + p(t)*y(t))
|
||
|
|
return "type7"
|
||
|
|
def check_nonlinear_2eq_order1(eq, func, func_coef):
|
||
|
|
t = list(list(eq[0].atoms(Derivative))[0].atoms(Symbol))[0]
|
||
|
|
f = Wild('f')
|
||
|
|
g = Wild('g')
|
||
|
|
u, v = symbols('u, v', cls=Dummy)
|
||
|
|
def check_type(x, y):
|
||
|
|
r1 = eq[0].match(t*diff(x(t),t) - x(t) + f)
|
||
|
|
r2 = eq[1].match(t*diff(y(t),t) - y(t) + g)
|
||
|
|
if not (r1 and r2):
|
||
|
|
r1 = eq[0].match(diff(x(t),t) - x(t)/t + f/t)
|
||
|
|
r2 = eq[1].match(diff(y(t),t) - y(t)/t + g/t)
|
||
|
|
if not (r1 and r2):
|
||
|
|
r1 = (-eq[0]).match(t*diff(x(t),t) - x(t) + f)
|
||
|
|
r2 = (-eq[1]).match(t*diff(y(t),t) - y(t) + g)
|
||
|
|
if not (r1 and r2):
|
||
|
|
r1 = (-eq[0]).match(diff(x(t),t) - x(t)/t + f/t)
|
||
|
|
r2 = (-eq[1]).match(diff(y(t),t) - y(t)/t + g/t)
|
||
|
|
if r1 and r2 and not (r1[f].subs(diff(x(t),t),u).subs(diff(y(t),t),v).has(t) \
|
||
|
|
or r2[g].subs(diff(x(t),t),u).subs(diff(y(t),t),v).has(t)):
|
||
|
|
return 'type5'
|
||
|
|
else:
|
||
|
|
return None
|
||
|
|
for func_ in func:
|
||
|
|
if isinstance(func_, list):
|
||
|
|
x = func[0][0].func
|
||
|
|
y = func[0][1].func
|
||
|
|
eq_type = check_type(x, y)
|
||
|
|
if not eq_type:
|
||
|
|
eq_type = check_type(y, x)
|
||
|
|
return eq_type
|
||
|
|
x = func[0].func
|
||
|
|
y = func[1].func
|
||
|
|
fc = func_coef
|
||
|
|
n = Wild('n', exclude=[x(t),y(t)])
|
||
|
|
f1 = Wild('f1', exclude=[v,t])
|
||
|
|
f2 = Wild('f2', exclude=[v,t])
|
||
|
|
g1 = Wild('g1', exclude=[u,t])
|
||
|
|
g2 = Wild('g2', exclude=[u,t])
|
||
|
|
for i in range(2):
|
||
|
|
eqs = 0
|
||
|
|
for terms in Add.make_args(eq[i]):
|
||
|
|
eqs += terms/fc[i,func[i],1]
|
||
|
|
eq[i] = eqs
|
||
|
|
r = eq[0].match(diff(x(t),t) - x(t)**n*f)
|
||
|
|
if r:
|
||
|
|
g = (diff(y(t),t) - eq[1])/r[f]
|
||
|
|
if r and not (g.has(x(t)) or g.subs(y(t),v).has(t) or r[f].subs(x(t),u).subs(y(t),v).has(t)):
|
||
|
|
return 'type1'
|
||
|
|
r = eq[0].match(diff(x(t),t) - exp(n*x(t))*f)
|
||
|
|
if r:
|
||
|
|
g = (diff(y(t),t) - eq[1])/r[f]
|
||
|
|
if r and not (g.has(x(t)) or g.subs(y(t),v).has(t) or r[f].subs(x(t),u).subs(y(t),v).has(t)):
|
||
|
|
return 'type2'
|
||
|
|
g = Wild('g')
|
||
|
|
r1 = eq[0].match(diff(x(t),t) - f)
|
||
|
|
r2 = eq[1].match(diff(y(t),t) - g)
|
||
|
|
if r1 and r2 and not (r1[f].subs(x(t),u).subs(y(t),v).has(t) or \
|
||
|
|
r2[g].subs(x(t),u).subs(y(t),v).has(t)):
|
||
|
|
return 'type3'
|
||
|
|
r1 = eq[0].match(diff(x(t),t) - f)
|
||
|
|
r2 = eq[1].match(diff(y(t),t) - g)
|
||
|
|
num, den = (
|
||
|
|
(r1[f].subs(x(t),u).subs(y(t),v))/
|
||
|
|
(r2[g].subs(x(t),u).subs(y(t),v))).as_numer_denom()
|
||
|
|
R1 = num.match(f1*g1)
|
||
|
|
R2 = den.match(f2*g2)
|
||
|
|
# phi = (r1[f].subs(x(t),u).subs(y(t),v))/num
|
||
|
|
if R1 and R2:
|
||
|
|
return 'type4'
|
||
|
|
return None
|
||
|
|
|
||
|
|
|
||
|
|
def check_nonlinear_2eq_order2(eq, func, func_coef):
|
||
|
|
return None
|
||
|
|
|
||
|
|
def check_nonlinear_3eq_order1(eq, func, func_coef):
|
||
|
|
x = func[0].func
|
||
|
|
y = func[1].func
|
||
|
|
z = func[2].func
|
||
|
|
fc = func_coef
|
||
|
|
t = list(list(eq[0].atoms(Derivative))[0].atoms(Symbol))[0]
|
||
|
|
u, v, w = symbols('u, v, w', cls=Dummy)
|
||
|
|
a = Wild('a', exclude=[x(t), y(t), z(t), t])
|
||
|
|
b = Wild('b', exclude=[x(t), y(t), z(t), t])
|
||
|
|
c = Wild('c', exclude=[x(t), y(t), z(t), t])
|
||
|
|
f = Wild('f')
|
||
|
|
F1 = Wild('F1')
|
||
|
|
F2 = Wild('F2')
|
||
|
|
F3 = Wild('F3')
|
||
|
|
for i in range(3):
|
||
|
|
eqs = 0
|
||
|
|
for terms in Add.make_args(eq[i]):
|
||
|
|
eqs += terms/fc[i,func[i],1]
|
||
|
|
eq[i] = eqs
|
||
|
|
r1 = eq[0].match(diff(x(t),t) - a*y(t)*z(t))
|
||
|
|
r2 = eq[1].match(diff(y(t),t) - b*z(t)*x(t))
|
||
|
|
r3 = eq[2].match(diff(z(t),t) - c*x(t)*y(t))
|
||
|
|
if r1 and r2 and r3:
|
||
|
|
num1, den1 = r1[a].as_numer_denom()
|
||
|
|
num2, den2 = r2[b].as_numer_denom()
|
||
|
|
num3, den3 = r3[c].as_numer_denom()
|
||
|
|
if solve([num1*u-den1*(v-w), num2*v-den2*(w-u), num3*w-den3*(u-v)],[u, v]):
|
||
|
|
return 'type1'
|
||
|
|
r = eq[0].match(diff(x(t),t) - y(t)*z(t)*f)
|
||
|
|
if r:
|
||
|
|
r1 = collect_const(r[f]).match(a*f)
|
||
|
|
r2 = ((diff(y(t),t) - eq[1])/r1[f]).match(b*z(t)*x(t))
|
||
|
|
r3 = ((diff(z(t),t) - eq[2])/r1[f]).match(c*x(t)*y(t))
|
||
|
|
if r1 and r2 and r3:
|
||
|
|
num1, den1 = r1[a].as_numer_denom()
|
||
|
|
num2, den2 = r2[b].as_numer_denom()
|
||
|
|
num3, den3 = r3[c].as_numer_denom()
|
||
|
|
if solve([num1*u-den1*(v-w), num2*v-den2*(w-u), num3*w-den3*(u-v)],[u, v]):
|
||
|
|
return 'type2'
|
||
|
|
r = eq[0].match(diff(x(t),t) - (F2-F3))
|
||
|
|
if r:
|
||
|
|
r1 = collect_const(r[F2]).match(c*F2)
|
||
|
|
r1.update(collect_const(r[F3]).match(b*F3))
|
||
|
|
if r1:
|
||
|
|
if eq[1].has(r1[F2]) and not eq[1].has(r1[F3]):
|
||
|
|
r1[F2], r1[F3] = r1[F3], r1[F2]
|
||
|
|
r1[c], r1[b] = -r1[b], -r1[c]
|
||
|
|
r2 = eq[1].match(diff(y(t),t) - a*r1[F3] + r1[c]*F1)
|
||
|
|
if r2:
|
||
|
|
r3 = (eq[2] == diff(z(t),t) - r1[b]*r2[F1] + r2[a]*r1[F2])
|
||
|
|
if r1 and r2 and r3:
|
||
|
|
return 'type3'
|
||
|
|
r = eq[0].match(diff(x(t),t) - z(t)*F2 + y(t)*F3)
|
||
|
|
if r:
|
||
|
|
r1 = collect_const(r[F2]).match(c*F2)
|
||
|
|
r1.update(collect_const(r[F3]).match(b*F3))
|
||
|
|
if r1:
|
||
|
|
if eq[1].has(r1[F2]) and not eq[1].has(r1[F3]):
|
||
|
|
r1[F2], r1[F3] = r1[F3], r1[F2]
|
||
|
|
r1[c], r1[b] = -r1[b], -r1[c]
|
||
|
|
r2 = (diff(y(t),t) - eq[1]).match(a*x(t)*r1[F3] - r1[c]*z(t)*F1)
|
||
|
|
if r2:
|
||
|
|
r3 = (diff(z(t),t) - eq[2] == r1[b]*y(t)*r2[F1] - r2[a]*x(t)*r1[F2])
|
||
|
|
if r1 and r2 and r3:
|
||
|
|
return 'type4'
|
||
|
|
r = (diff(x(t),t) - eq[0]).match(x(t)*(F2 - F3))
|
||
|
|
if r:
|
||
|
|
r1 = collect_const(r[F2]).match(c*F2)
|
||
|
|
r1.update(collect_const(r[F3]).match(b*F3))
|
||
|
|
if r1:
|
||
|
|
if eq[1].has(r1[F2]) and not eq[1].has(r1[F3]):
|
||
|
|
r1[F2], r1[F3] = r1[F3], r1[F2]
|
||
|
|
r1[c], r1[b] = -r1[b], -r1[c]
|
||
|
|
r2 = (diff(y(t),t) - eq[1]).match(y(t)*(a*r1[F3] - r1[c]*F1))
|
||
|
|
if r2:
|
||
|
|
r3 = (diff(z(t),t) - eq[2] == z(t)*(r1[b]*r2[F1] - r2[a]*r1[F2]))
|
||
|
|
if r1 and r2 and r3:
|
||
|
|
return 'type5'
|
||
|
|
return None
|
||
|
|
|
||
|
|
|
||
|
|
def check_nonlinear_3eq_order2(eq, func, func_coef):
|
||
|
|
return None
|
||
|
|
|
||
|
|
|
||
|
|
@vectorize(0)
|
||
|
|
def odesimp(ode, eq, func, hint):
|
||
|
|
r"""
|
||
|
|
Simplifies solutions of ODEs, including trying to solve for ``func`` and
|
||
|
|
running :py:meth:`~sympy.solvers.ode.constantsimp`.
|
||
|
|
|
||
|
|
It may use knowledge of the type of solution that the hint returns to
|
||
|
|
apply additional simplifications.
|
||
|
|
|
||
|
|
It also attempts to integrate any :py:class:`~sympy.integrals.integrals.Integral`\s
|
||
|
|
in the expression, if the hint is not an ``_Integral`` hint.
|
||
|
|
|
||
|
|
This function should have no effect on expressions returned by
|
||
|
|
:py:meth:`~sympy.solvers.ode.dsolve`, as
|
||
|
|
:py:meth:`~sympy.solvers.ode.dsolve` already calls
|
||
|
|
:py:meth:`~sympy.solvers.ode.ode.odesimp`, but the individual hint functions
|
||
|
|
do not call :py:meth:`~sympy.solvers.ode.ode.odesimp` (because the
|
||
|
|
:py:meth:`~sympy.solvers.ode.dsolve` wrapper does). Therefore, this
|
||
|
|
function is designed for mainly internal use.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
>>> from sympy import sin, symbols, dsolve, pprint, Function
|
||
|
|
>>> from sympy.solvers.ode.ode import odesimp
|
||
|
|
>>> x, u2, C1= symbols('x,u2,C1')
|
||
|
|
>>> f = Function('f')
|
||
|
|
|
||
|
|
>>> eq = dsolve(x*f(x).diff(x) - f(x) - x*sin(f(x)/x), f(x),
|
||
|
|
... hint='1st_homogeneous_coeff_subs_indep_div_dep_Integral',
|
||
|
|
... simplify=False)
|
||
|
|
>>> pprint(eq, wrap_line=False)
|
||
|
|
x
|
||
|
|
----
|
||
|
|
f(x)
|
||
|
|
/
|
||
|
|
|
|
||
|
|
| / 1 \
|
||
|
|
| -|u1 + -------|
|
||
|
|
| | /1 \|
|
||
|
|
| | sin|--||
|
||
|
|
| \ \u1//
|
||
|
|
log(f(x)) = log(C1) + | ---------------- d(u1)
|
||
|
|
| 2
|
||
|
|
| u1
|
||
|
|
|
|
||
|
|
/
|
||
|
|
|
||
|
|
>>> pprint(odesimp(eq, f(x), 1, {C1},
|
||
|
|
... hint='1st_homogeneous_coeff_subs_indep_div_dep'
|
||
|
|
... )) #doctest: +SKIP
|
||
|
|
x
|
||
|
|
--------- = C1
|
||
|
|
/f(x)\
|
||
|
|
tan|----|
|
||
|
|
\2*x /
|
||
|
|
|
||
|
|
"""
|
||
|
|
x = func.args[0]
|
||
|
|
f = func.func
|
||
|
|
C1 = get_numbered_constants(eq, num=1)
|
||
|
|
constants = eq.free_symbols - ode.free_symbols
|
||
|
|
|
||
|
|
# First, integrate if the hint allows it.
|
||
|
|
eq = _handle_Integral(eq, func, hint)
|
||
|
|
if hint.startswith("nth_linear_euler_eq_nonhomogeneous"):
|
||
|
|
eq = simplify(eq)
|
||
|
|
if not isinstance(eq, Equality):
|
||
|
|
raise TypeError("eq should be an instance of Equality")
|
||
|
|
|
||
|
|
# Second, clean up the arbitrary constants.
|
||
|
|
# Right now, nth linear hints can put as many as 2*order constants in an
|
||
|
|
# expression. If that number grows with another hint, the third argument
|
||
|
|
# here should be raised accordingly, or constantsimp() rewritten to handle
|
||
|
|
# an arbitrary number of constants.
|
||
|
|
eq = constantsimp(eq, constants)
|
||
|
|
|
||
|
|
# Lastly, now that we have cleaned up the expression, try solving for func.
|
||
|
|
# When CRootOf is implemented in solve(), we will want to return a CRootOf
|
||
|
|
# every time instead of an Equality.
|
||
|
|
|
||
|
|
# Get the f(x) on the left if possible.
|
||
|
|
if eq.rhs == func and not eq.lhs.has(func):
|
||
|
|
eq = [Eq(eq.rhs, eq.lhs)]
|
||
|
|
|
||
|
|
# make sure we are working with lists of solutions in simplified form.
|
||
|
|
if eq.lhs == func and not eq.rhs.has(func):
|
||
|
|
# The solution is already solved
|
||
|
|
eq = [eq]
|
||
|
|
|
||
|
|
else:
|
||
|
|
# The solution is not solved, so try to solve it
|
||
|
|
try:
|
||
|
|
floats = any(i.is_Float for i in eq.atoms(Number))
|
||
|
|
eqsol = solve(eq, func, force=True, rational=False if floats else None)
|
||
|
|
if not eqsol:
|
||
|
|
raise NotImplementedError
|
||
|
|
except (NotImplementedError, PolynomialError):
|
||
|
|
eq = [eq]
|
||
|
|
else:
|
||
|
|
def _expand(expr):
|
||
|
|
numer, denom = expr.as_numer_denom()
|
||
|
|
|
||
|
|
if denom.is_Add:
|
||
|
|
return expr
|
||
|
|
else:
|
||
|
|
return powsimp(expr.expand(), combine='exp', deep=True)
|
||
|
|
|
||
|
|
# XXX: the rest of odesimp() expects each ``t`` to be in a
|
||
|
|
# specific normal form: rational expression with numerator
|
||
|
|
# expanded, but with combined exponential functions (at
|
||
|
|
# least in this setup all tests pass).
|
||
|
|
eq = [Eq(f(x), _expand(t)) for t in eqsol]
|
||
|
|
|
||
|
|
# special simplification of the lhs.
|
||
|
|
if hint.startswith("1st_homogeneous_coeff"):
|
||
|
|
for j, eqi in enumerate(eq):
|
||
|
|
newi = logcombine(eqi, force=True)
|
||
|
|
if isinstance(newi.lhs, log) and newi.rhs == 0:
|
||
|
|
newi = Eq(newi.lhs.args[0]/C1, C1)
|
||
|
|
eq[j] = newi
|
||
|
|
|
||
|
|
# We cleaned up the constants before solving to help the solve engine with
|
||
|
|
# a simpler expression, but the solved expression could have introduced
|
||
|
|
# things like -C1, so rerun constantsimp() one last time before returning.
|
||
|
|
for i, eqi in enumerate(eq):
|
||
|
|
eq[i] = constantsimp(eqi, constants)
|
||
|
|
eq[i] = constant_renumber(eq[i], ode.free_symbols)
|
||
|
|
|
||
|
|
# If there is only 1 solution, return it;
|
||
|
|
# otherwise return the list of solutions.
|
||
|
|
if len(eq) == 1:
|
||
|
|
eq = eq[0]
|
||
|
|
return eq
|
||
|
|
|
||
|
|
|
||
|
|
def ode_sol_simplicity(sol, func, trysolving=True):
|
||
|
|
r"""
|
||
|
|
Returns an extended integer representing how simple a solution to an ODE
|
||
|
|
is.
|
||
|
|
|
||
|
|
The following things are considered, in order from most simple to least:
|
||
|
|
|
||
|
|
- ``sol`` is solved for ``func``.
|
||
|
|
- ``sol`` is not solved for ``func``, but can be if passed to solve (e.g.,
|
||
|
|
a solution returned by ``dsolve(ode, func, simplify=False``).
|
||
|
|
- If ``sol`` is not solved for ``func``, then base the result on the
|
||
|
|
length of ``sol``, as computed by ``len(str(sol))``.
|
||
|
|
- If ``sol`` has any unevaluated :py:class:`~sympy.integrals.integrals.Integral`\s,
|
||
|
|
this will automatically be considered less simple than any of the above.
|
||
|
|
|
||
|
|
This function returns an integer such that if solution A is simpler than
|
||
|
|
solution B by above metric, then ``ode_sol_simplicity(sola, func) <
|
||
|
|
ode_sol_simplicity(solb, func)``.
|
||
|
|
|
||
|
|
Currently, the following are the numbers returned, but if the heuristic is
|
||
|
|
ever improved, this may change. Only the ordering is guaranteed.
|
||
|
|
|
||
|
|
+----------------------------------------------+-------------------+
|
||
|
|
| Simplicity | Return |
|
||
|
|
+==============================================+===================+
|
||
|
|
| ``sol`` solved for ``func`` | ``-2`` |
|
||
|
|
+----------------------------------------------+-------------------+
|
||
|
|
| ``sol`` not solved for ``func`` but can be | ``-1`` |
|
||
|
|
+----------------------------------------------+-------------------+
|
||
|
|
| ``sol`` is not solved nor solvable for | ``len(str(sol))`` |
|
||
|
|
| ``func`` | |
|
||
|
|
+----------------------------------------------+-------------------+
|
||
|
|
| ``sol`` contains an | ``oo`` |
|
||
|
|
| :obj:`~sympy.integrals.integrals.Integral` | |
|
||
|
|
+----------------------------------------------+-------------------+
|
||
|
|
|
||
|
|
``oo`` here means the SymPy infinity, which should compare greater than
|
||
|
|
any integer.
|
||
|
|
|
||
|
|
If you already know :py:meth:`~sympy.solvers.solvers.solve` cannot solve
|
||
|
|
``sol``, you can use ``trysolving=False`` to skip that step, which is the
|
||
|
|
only potentially slow step. For example,
|
||
|
|
:py:meth:`~sympy.solvers.ode.dsolve` with the ``simplify=False`` flag
|
||
|
|
should do this.
|
||
|
|
|
||
|
|
If ``sol`` is a list of solutions, if the worst solution in the list
|
||
|
|
returns ``oo`` it returns that, otherwise it returns ``len(str(sol))``,
|
||
|
|
that is, the length of the string representation of the whole list.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
This function is designed to be passed to ``min`` as the key argument,
|
||
|
|
such as ``min(listofsolutions, key=lambda i: ode_sol_simplicity(i,
|
||
|
|
f(x)))``.
|
||
|
|
|
||
|
|
>>> from sympy import symbols, Function, Eq, tan, Integral
|
||
|
|
>>> from sympy.solvers.ode.ode import ode_sol_simplicity
|
||
|
|
>>> x, C1, C2 = symbols('x, C1, C2')
|
||
|
|
>>> f = Function('f')
|
||
|
|
|
||
|
|
>>> ode_sol_simplicity(Eq(f(x), C1*x**2), f(x))
|
||
|
|
-2
|
||
|
|
>>> ode_sol_simplicity(Eq(x**2 + f(x), C1), f(x))
|
||
|
|
-1
|
||
|
|
>>> ode_sol_simplicity(Eq(f(x), C1*Integral(2*x, x)), f(x))
|
||
|
|
oo
|
||
|
|
>>> eq1 = Eq(f(x)/tan(f(x)/(2*x)), C1)
|
||
|
|
>>> eq2 = Eq(f(x)/tan(f(x)/(2*x) + f(x)), C2)
|
||
|
|
>>> [ode_sol_simplicity(eq, f(x)) for eq in [eq1, eq2]]
|
||
|
|
[28, 35]
|
||
|
|
>>> min([eq1, eq2], key=lambda i: ode_sol_simplicity(i, f(x)))
|
||
|
|
Eq(f(x)/tan(f(x)/(2*x)), C1)
|
||
|
|
|
||
|
|
"""
|
||
|
|
# TODO: if two solutions are solved for f(x), we still want to be
|
||
|
|
# able to get the simpler of the two
|
||
|
|
|
||
|
|
# See the docstring for the coercion rules. We check easier (faster)
|
||
|
|
# things here first, to save time.
|
||
|
|
|
||
|
|
if iterable(sol):
|
||
|
|
# See if there are Integrals
|
||
|
|
for i in sol:
|
||
|
|
if ode_sol_simplicity(i, func, trysolving=trysolving) == oo:
|
||
|
|
return oo
|
||
|
|
|
||
|
|
return len(str(sol))
|
||
|
|
|
||
|
|
if sol.has(Integral):
|
||
|
|
return oo
|
||
|
|
|
||
|
|
# Next, try to solve for func. This code will change slightly when CRootOf
|
||
|
|
# is implemented in solve(). Probably a CRootOf solution should fall
|
||
|
|
# somewhere between a normal solution and an unsolvable expression.
|
||
|
|
|
||
|
|
# First, see if they are already solved
|
||
|
|
if sol.lhs == func and not sol.rhs.has(func) or \
|
||
|
|
sol.rhs == func and not sol.lhs.has(func):
|
||
|
|
return -2
|
||
|
|
# We are not so lucky, try solving manually
|
||
|
|
if trysolving:
|
||
|
|
try:
|
||
|
|
sols = solve(sol, func)
|
||
|
|
if not sols:
|
||
|
|
raise NotImplementedError
|
||
|
|
except NotImplementedError:
|
||
|
|
pass
|
||
|
|
else:
|
||
|
|
return -1
|
||
|
|
|
||
|
|
# Finally, a naive computation based on the length of the string version
|
||
|
|
# of the expression. This may favor combined fractions because they
|
||
|
|
# will not have duplicate denominators, and may slightly favor expressions
|
||
|
|
# with fewer additions and subtractions, as those are separated by spaces
|
||
|
|
# by the printer.
|
||
|
|
|
||
|
|
# Additional ideas for simplicity heuristics are welcome, like maybe
|
||
|
|
# checking if a equation has a larger domain, or if constantsimp has
|
||
|
|
# introduced arbitrary constants numbered higher than the order of a
|
||
|
|
# given ODE that sol is a solution of.
|
||
|
|
return len(str(sol))
|
||
|
|
|
||
|
|
|
||
|
|
def _extract_funcs(eqs):
|
||
|
|
funcs = []
|
||
|
|
for eq in eqs:
|
||
|
|
derivs = [node for node in preorder_traversal(eq) if isinstance(node, Derivative)]
|
||
|
|
func = []
|
||
|
|
for d in derivs:
|
||
|
|
func += list(d.atoms(AppliedUndef))
|
||
|
|
for func_ in func:
|
||
|
|
funcs.append(func_)
|
||
|
|
funcs = list(uniq(funcs))
|
||
|
|
|
||
|
|
return funcs
|
||
|
|
|
||
|
|
|
||
|
|
def _get_constant_subexpressions(expr, Cs):
|
||
|
|
Cs = set(Cs)
|
||
|
|
Ces = []
|
||
|
|
def _recursive_walk(expr):
|
||
|
|
expr_syms = expr.free_symbols
|
||
|
|
if expr_syms and expr_syms.issubset(Cs):
|
||
|
|
Ces.append(expr)
|
||
|
|
else:
|
||
|
|
if expr.func == exp:
|
||
|
|
expr = expr.expand(mul=True)
|
||
|
|
if expr.func in (Add, Mul):
|
||
|
|
d = sift(expr.args, lambda i : i.free_symbols.issubset(Cs))
|
||
|
|
if len(d[True]) > 1:
|
||
|
|
x = expr.func(*d[True])
|
||
|
|
if not x.is_number:
|
||
|
|
Ces.append(x)
|
||
|
|
elif isinstance(expr, Integral):
|
||
|
|
if expr.free_symbols.issubset(Cs) and \
|
||
|
|
all(len(x) == 3 for x in expr.limits):
|
||
|
|
Ces.append(expr)
|
||
|
|
for i in expr.args:
|
||
|
|
_recursive_walk(i)
|
||
|
|
return
|
||
|
|
_recursive_walk(expr)
|
||
|
|
return Ces
|
||
|
|
|
||
|
|
def __remove_linear_redundancies(expr, Cs):
|
||
|
|
cnts = {i: expr.count(i) for i in Cs}
|
||
|
|
Cs = [i for i in Cs if cnts[i] > 0]
|
||
|
|
|
||
|
|
def _linear(expr):
|
||
|
|
if isinstance(expr, Add):
|
||
|
|
xs = [i for i in Cs if expr.count(i)==cnts[i] \
|
||
|
|
and 0 == expr.diff(i, 2)]
|
||
|
|
d = {}
|
||
|
|
for x in xs:
|
||
|
|
y = expr.diff(x)
|
||
|
|
if y not in d:
|
||
|
|
d[y]=[]
|
||
|
|
d[y].append(x)
|
||
|
|
for y in d:
|
||
|
|
if len(d[y]) > 1:
|
||
|
|
d[y].sort(key=str)
|
||
|
|
for x in d[y][1:]:
|
||
|
|
expr = expr.subs(x, 0)
|
||
|
|
return expr
|
||
|
|
|
||
|
|
def _recursive_walk(expr):
|
||
|
|
if len(expr.args) != 0:
|
||
|
|
expr = expr.func(*[_recursive_walk(i) for i in expr.args])
|
||
|
|
expr = _linear(expr)
|
||
|
|
return expr
|
||
|
|
|
||
|
|
if isinstance(expr, Equality):
|
||
|
|
lhs, rhs = [_recursive_walk(i) for i in expr.args]
|
||
|
|
f = lambda i: isinstance(i, Number) or i in Cs
|
||
|
|
if isinstance(lhs, Symbol) and lhs in Cs:
|
||
|
|
rhs, lhs = lhs, rhs
|
||
|
|
if lhs.func in (Add, Symbol) and rhs.func in (Add, Symbol):
|
||
|
|
dlhs = sift([lhs] if isinstance(lhs, AtomicExpr) else lhs.args, f)
|
||
|
|
drhs = sift([rhs] if isinstance(rhs, AtomicExpr) else rhs.args, f)
|
||
|
|
for i in [True, False]:
|
||
|
|
for hs in [dlhs, drhs]:
|
||
|
|
if i not in hs:
|
||
|
|
hs[i] = [0]
|
||
|
|
# this calculation can be simplified
|
||
|
|
lhs = Add(*dlhs[False]) - Add(*drhs[False])
|
||
|
|
rhs = Add(*drhs[True]) - Add(*dlhs[True])
|
||
|
|
elif lhs.func in (Mul, Symbol) and rhs.func in (Mul, Symbol):
|
||
|
|
dlhs = sift([lhs] if isinstance(lhs, AtomicExpr) else lhs.args, f)
|
||
|
|
if True in dlhs:
|
||
|
|
if False not in dlhs:
|
||
|
|
dlhs[False] = [1]
|
||
|
|
lhs = Mul(*dlhs[False])
|
||
|
|
rhs = rhs/Mul(*dlhs[True])
|
||
|
|
return Eq(lhs, rhs)
|
||
|
|
else:
|
||
|
|
return _recursive_walk(expr)
|
||
|
|
|
||
|
|
@vectorize(0)
|
||
|
|
def constantsimp(expr, constants):
|
||
|
|
r"""
|
||
|
|
Simplifies an expression with arbitrary constants in it.
|
||
|
|
|
||
|
|
This function is written specifically to work with
|
||
|
|
:py:meth:`~sympy.solvers.ode.dsolve`, and is not intended for general use.
|
||
|
|
|
||
|
|
Simplification is done by "absorbing" the arbitrary constants into other
|
||
|
|
arbitrary constants, numbers, and symbols that they are not independent
|
||
|
|
of.
|
||
|
|
|
||
|
|
The symbols must all have the same name with numbers after it, for
|
||
|
|
example, ``C1``, ``C2``, ``C3``. The ``symbolname`` here would be
|
||
|
|
'``C``', the ``startnumber`` would be 1, and the ``endnumber`` would be 3.
|
||
|
|
If the arbitrary constants are independent of the variable ``x``, then the
|
||
|
|
independent symbol would be ``x``. There is no need to specify the
|
||
|
|
dependent function, such as ``f(x)``, because it already has the
|
||
|
|
independent symbol, ``x``, in it.
|
||
|
|
|
||
|
|
Because terms are "absorbed" into arbitrary constants and because
|
||
|
|
constants are renumbered after simplifying, the arbitrary constants in
|
||
|
|
expr are not necessarily equal to the ones of the same name in the
|
||
|
|
returned result.
|
||
|
|
|
||
|
|
If two or more arbitrary constants are added, multiplied, or raised to the
|
||
|
|
power of each other, they are first absorbed together into a single
|
||
|
|
arbitrary constant. Then the new constant is combined into other terms if
|
||
|
|
necessary.
|
||
|
|
|
||
|
|
Absorption of constants is done with limited assistance:
|
||
|
|
|
||
|
|
1. terms of :py:class:`~sympy.core.add.Add`\s are collected to try join
|
||
|
|
constants so `e^x (C_1 \cos(x) + C_2 \cos(x))` will simplify to `e^x
|
||
|
|
C_1 \cos(x)`;
|
||
|
|
|
||
|
|
2. powers with exponents that are :py:class:`~sympy.core.add.Add`\s are
|
||
|
|
expanded so `e^{C_1 + x}` will be simplified to `C_1 e^x`.
|
||
|
|
|
||
|
|
Use :py:meth:`~sympy.solvers.ode.ode.constant_renumber` to renumber constants
|
||
|
|
after simplification or else arbitrary numbers on constants may appear,
|
||
|
|
e.g. `C_1 + C_3 x`.
|
||
|
|
|
||
|
|
In rare cases, a single constant can be "simplified" into two constants.
|
||
|
|
Every differential equation solution should have as many arbitrary
|
||
|
|
constants as the order of the differential equation. The result here will
|
||
|
|
be technically correct, but it may, for example, have `C_1` and `C_2` in
|
||
|
|
an expression, when `C_1` is actually equal to `C_2`. Use your discretion
|
||
|
|
in such situations, and also take advantage of the ability to use hints in
|
||
|
|
:py:meth:`~sympy.solvers.ode.dsolve`.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
>>> from sympy import symbols
|
||
|
|
>>> from sympy.solvers.ode.ode import constantsimp
|
||
|
|
>>> C1, C2, C3, x, y = symbols('C1, C2, C3, x, y')
|
||
|
|
>>> constantsimp(2*C1*x, {C1, C2, C3})
|
||
|
|
C1*x
|
||
|
|
>>> constantsimp(C1 + 2 + x, {C1, C2, C3})
|
||
|
|
C1 + x
|
||
|
|
>>> constantsimp(C1*C2 + 2 + C2 + C3*x, {C1, C2, C3})
|
||
|
|
C1 + C3*x
|
||
|
|
|
||
|
|
"""
|
||
|
|
# This function works recursively. The idea is that, for Mul,
|
||
|
|
# Add, Pow, and Function, if the class has a constant in it, then
|
||
|
|
# we can simplify it, which we do by recursing down and
|
||
|
|
# simplifying up. Otherwise, we can skip that part of the
|
||
|
|
# expression.
|
||
|
|
|
||
|
|
Cs = constants
|
||
|
|
|
||
|
|
orig_expr = expr
|
||
|
|
|
||
|
|
constant_subexprs = _get_constant_subexpressions(expr, Cs)
|
||
|
|
for xe in constant_subexprs:
|
||
|
|
xes = list(xe.free_symbols)
|
||
|
|
if not xes:
|
||
|
|
continue
|
||
|
|
if all(expr.count(c) == xe.count(c) for c in xes):
|
||
|
|
xes.sort(key=str)
|
||
|
|
expr = expr.subs(xe, xes[0])
|
||
|
|
|
||
|
|
# try to perform common sub-expression elimination of constant terms
|
||
|
|
try:
|
||
|
|
commons, rexpr = cse(expr)
|
||
|
|
commons.reverse()
|
||
|
|
rexpr = rexpr[0]
|
||
|
|
for s in commons:
|
||
|
|
cs = list(s[1].atoms(Symbol))
|
||
|
|
if len(cs) == 1 and cs[0] in Cs and \
|
||
|
|
cs[0] not in rexpr.atoms(Symbol) and \
|
||
|
|
not any(cs[0] in ex for ex in commons if ex != s):
|
||
|
|
rexpr = rexpr.subs(s[0], cs[0])
|
||
|
|
else:
|
||
|
|
rexpr = rexpr.subs(*s)
|
||
|
|
expr = rexpr
|
||
|
|
except IndexError:
|
||
|
|
pass
|
||
|
|
expr = __remove_linear_redundancies(expr, Cs)
|
||
|
|
|
||
|
|
def _conditional_term_factoring(expr):
|
||
|
|
new_expr = terms_gcd(expr, clear=False, deep=True, expand=False)
|
||
|
|
|
||
|
|
# we do not want to factor exponentials, so handle this separately
|
||
|
|
if new_expr.is_Mul:
|
||
|
|
infac = False
|
||
|
|
asfac = False
|
||
|
|
for m in new_expr.args:
|
||
|
|
if isinstance(m, exp):
|
||
|
|
asfac = True
|
||
|
|
elif m.is_Add:
|
||
|
|
infac = any(isinstance(fi, exp) for t in m.args
|
||
|
|
for fi in Mul.make_args(t))
|
||
|
|
if asfac and infac:
|
||
|
|
new_expr = expr
|
||
|
|
break
|
||
|
|
return new_expr
|
||
|
|
|
||
|
|
expr = _conditional_term_factoring(expr)
|
||
|
|
|
||
|
|
# call recursively if more simplification is possible
|
||
|
|
if orig_expr != expr:
|
||
|
|
return constantsimp(expr, Cs)
|
||
|
|
return expr
|
||
|
|
|
||
|
|
|
||
|
|
def constant_renumber(expr, variables=None, newconstants=None):
|
||
|
|
r"""
|
||
|
|
Renumber arbitrary constants in ``expr`` to use the symbol names as given
|
||
|
|
in ``newconstants``. In the process, this reorders expression terms in a
|
||
|
|
standard way.
|
||
|
|
|
||
|
|
If ``newconstants`` is not provided then the new constant names will be
|
||
|
|
``C1``, ``C2`` etc. Otherwise ``newconstants`` should be an iterable
|
||
|
|
giving the new symbols to use for the constants in order.
|
||
|
|
|
||
|
|
The ``variables`` argument is a list of non-constant symbols. All other
|
||
|
|
free symbols found in ``expr`` are assumed to be constants and will be
|
||
|
|
renumbered. If ``variables`` is not given then any numbered symbol
|
||
|
|
beginning with ``C`` (e.g. ``C1``) is assumed to be a constant.
|
||
|
|
|
||
|
|
Symbols are renumbered based on ``.sort_key()``, so they should be
|
||
|
|
numbered roughly in the order that they appear in the final, printed
|
||
|
|
expression. Note that this ordering is based in part on hashes, so it can
|
||
|
|
produce different results on different machines.
|
||
|
|
|
||
|
|
The structure of this function is very similar to that of
|
||
|
|
:py:meth:`~sympy.solvers.ode.constantsimp`.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
>>> from sympy import symbols
|
||
|
|
>>> from sympy.solvers.ode.ode import constant_renumber
|
||
|
|
>>> x, C1, C2, C3 = symbols('x,C1:4')
|
||
|
|
>>> expr = C3 + C2*x + C1*x**2
|
||
|
|
>>> expr
|
||
|
|
C1*x**2 + C2*x + C3
|
||
|
|
>>> constant_renumber(expr)
|
||
|
|
C1 + C2*x + C3*x**2
|
||
|
|
|
||
|
|
The ``variables`` argument specifies which are constants so that the
|
||
|
|
other symbols will not be renumbered:
|
||
|
|
|
||
|
|
>>> constant_renumber(expr, [C1, x])
|
||
|
|
C1*x**2 + C2 + C3*x
|
||
|
|
|
||
|
|
The ``newconstants`` argument is used to specify what symbols to use when
|
||
|
|
replacing the constants:
|
||
|
|
|
||
|
|
>>> constant_renumber(expr, [x], newconstants=symbols('E1:4'))
|
||
|
|
E1 + E2*x + E3*x**2
|
||
|
|
|
||
|
|
"""
|
||
|
|
|
||
|
|
# System of expressions
|
||
|
|
if isinstance(expr, (set, list, tuple)):
|
||
|
|
return type(expr)(constant_renumber(Tuple(*expr),
|
||
|
|
variables=variables, newconstants=newconstants))
|
||
|
|
|
||
|
|
# Symbols in solution but not ODE are constants
|
||
|
|
if variables is not None:
|
||
|
|
variables = set(variables)
|
||
|
|
free_symbols = expr.free_symbols
|
||
|
|
constantsymbols = list(free_symbols - variables)
|
||
|
|
# Any Cn is a constant...
|
||
|
|
else:
|
||
|
|
variables = set()
|
||
|
|
isconstant = lambda s: s.startswith('C') and s[1:].isdigit()
|
||
|
|
constantsymbols = [sym for sym in expr.free_symbols if isconstant(sym.name)]
|
||
|
|
|
||
|
|
# Find new constants checking that they aren't already in the ODE
|
||
|
|
if newconstants is None:
|
||
|
|
iter_constants = numbered_symbols(start=1, prefix='C', exclude=variables)
|
||
|
|
else:
|
||
|
|
iter_constants = (sym for sym in newconstants if sym not in variables)
|
||
|
|
|
||
|
|
constants_found = []
|
||
|
|
|
||
|
|
# make a mapping to send all constantsymbols to S.One and use
|
||
|
|
# that to make sure that term ordering is not dependent on
|
||
|
|
# the indexed value of C
|
||
|
|
C_1 = [(ci, S.One) for ci in constantsymbols]
|
||
|
|
sort_key=lambda arg: default_sort_key(arg.subs(C_1))
|
||
|
|
|
||
|
|
def _constant_renumber(expr):
|
||
|
|
r"""
|
||
|
|
We need to have an internal recursive function
|
||
|
|
"""
|
||
|
|
|
||
|
|
# For system of expressions
|
||
|
|
if isinstance(expr, Tuple):
|
||
|
|
renumbered = [_constant_renumber(e) for e in expr]
|
||
|
|
return Tuple(*renumbered)
|
||
|
|
|
||
|
|
if isinstance(expr, Equality):
|
||
|
|
return Eq(
|
||
|
|
_constant_renumber(expr.lhs),
|
||
|
|
_constant_renumber(expr.rhs))
|
||
|
|
|
||
|
|
if type(expr) not in (Mul, Add, Pow) and not expr.is_Function and \
|
||
|
|
not expr.has(*constantsymbols):
|
||
|
|
# Base case, as above. Hope there aren't constants inside
|
||
|
|
# of some other class, because they won't be renumbered.
|
||
|
|
return expr
|
||
|
|
elif expr.is_Piecewise:
|
||
|
|
return expr
|
||
|
|
elif expr in constantsymbols:
|
||
|
|
if expr not in constants_found:
|
||
|
|
constants_found.append(expr)
|
||
|
|
return expr
|
||
|
|
elif expr.is_Function or expr.is_Pow:
|
||
|
|
return expr.func(
|
||
|
|
*[_constant_renumber(x) for x in expr.args])
|
||
|
|
else:
|
||
|
|
sortedargs = list(expr.args)
|
||
|
|
sortedargs.sort(key=sort_key)
|
||
|
|
return expr.func(*[_constant_renumber(x) for x in sortedargs])
|
||
|
|
expr = _constant_renumber(expr)
|
||
|
|
|
||
|
|
# Don't renumber symbols present in the ODE.
|
||
|
|
constants_found = [c for c in constants_found if c not in variables]
|
||
|
|
|
||
|
|
# Renumbering happens here
|
||
|
|
subs_dict = {var: cons for var, cons in zip(constants_found, iter_constants)}
|
||
|
|
expr = expr.subs(subs_dict, simultaneous=True)
|
||
|
|
|
||
|
|
return expr
|
||
|
|
|
||
|
|
|
||
|
|
def _handle_Integral(expr, func, hint):
|
||
|
|
r"""
|
||
|
|
Converts a solution with Integrals in it into an actual solution.
|
||
|
|
|
||
|
|
For most hints, this simply runs ``expr.doit()``.
|
||
|
|
|
||
|
|
"""
|
||
|
|
if hint == "nth_linear_constant_coeff_homogeneous":
|
||
|
|
sol = expr
|
||
|
|
elif not hint.endswith("_Integral"):
|
||
|
|
sol = expr.doit()
|
||
|
|
else:
|
||
|
|
sol = expr
|
||
|
|
return sol
|
||
|
|
|
||
|
|
|
||
|
|
# XXX: Should this function maybe go somewhere else?
|
||
|
|
|
||
|
|
|
||
|
|
def homogeneous_order(eq, *symbols):
|
||
|
|
r"""
|
||
|
|
Returns the order `n` if `g` is homogeneous and ``None`` if it is not
|
||
|
|
homogeneous.
|
||
|
|
|
||
|
|
Determines if a function is homogeneous and if so of what order. A
|
||
|
|
function `f(x, y, \cdots)` is homogeneous of order `n` if `f(t x, t y,
|
||
|
|
\cdots) = t^n f(x, y, \cdots)`.
|
||
|
|
|
||
|
|
If the function is of two variables, `F(x, y)`, then `f` being homogeneous
|
||
|
|
of any order is equivalent to being able to rewrite `F(x, y)` as `G(x/y)`
|
||
|
|
or `H(y/x)`. This fact is used to solve 1st order ordinary differential
|
||
|
|
equations whose coefficients are homogeneous of the same order (see the
|
||
|
|
docstrings of
|
||
|
|
:obj:`~sympy.solvers.ode.single.HomogeneousCoeffSubsDepDivIndep` and
|
||
|
|
:obj:`~sympy.solvers.ode.single.HomogeneousCoeffSubsIndepDivDep`).
|
||
|
|
|
||
|
|
Symbols can be functions, but every argument of the function must be a
|
||
|
|
symbol, and the arguments of the function that appear in the expression
|
||
|
|
must match those given in the list of symbols. If a declared function
|
||
|
|
appears with different arguments than given in the list of symbols,
|
||
|
|
``None`` is returned.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
>>> from sympy import Function, homogeneous_order, sqrt
|
||
|
|
>>> from sympy.abc import x, y
|
||
|
|
>>> f = Function('f')
|
||
|
|
>>> homogeneous_order(f(x), f(x)) is None
|
||
|
|
True
|
||
|
|
>>> homogeneous_order(f(x,y), f(y, x), x, y) is None
|
||
|
|
True
|
||
|
|
>>> homogeneous_order(f(x), f(x), x)
|
||
|
|
1
|
||
|
|
>>> homogeneous_order(x**2*f(x)/sqrt(x**2+f(x)**2), x, f(x))
|
||
|
|
2
|
||
|
|
>>> homogeneous_order(x**2+f(x), x, f(x)) is None
|
||
|
|
True
|
||
|
|
|
||
|
|
"""
|
||
|
|
|
||
|
|
if not symbols:
|
||
|
|
raise ValueError("homogeneous_order: no symbols were given.")
|
||
|
|
symset = set(symbols)
|
||
|
|
eq = sympify(eq)
|
||
|
|
|
||
|
|
# The following are not supported
|
||
|
|
if eq.has(Order, Derivative):
|
||
|
|
return None
|
||
|
|
|
||
|
|
# These are all constants
|
||
|
|
if (eq.is_Number or
|
||
|
|
eq.is_NumberSymbol or
|
||
|
|
eq.is_number
|
||
|
|
):
|
||
|
|
return S.Zero
|
||
|
|
|
||
|
|
# Replace all functions with dummy variables
|
||
|
|
dum = numbered_symbols(prefix='d', cls=Dummy)
|
||
|
|
newsyms = set()
|
||
|
|
for i in [j for j in symset if getattr(j, 'is_Function')]:
|
||
|
|
iargs = set(i.args)
|
||
|
|
if iargs.difference(symset):
|
||
|
|
return None
|
||
|
|
else:
|
||
|
|
dummyvar = next(dum)
|
||
|
|
eq = eq.subs(i, dummyvar)
|
||
|
|
symset.remove(i)
|
||
|
|
newsyms.add(dummyvar)
|
||
|
|
symset.update(newsyms)
|
||
|
|
|
||
|
|
if not eq.free_symbols & symset:
|
||
|
|
return None
|
||
|
|
|
||
|
|
# assuming order of a nested function can only be equal to zero
|
||
|
|
if isinstance(eq, Function):
|
||
|
|
return None if homogeneous_order(
|
||
|
|
eq.args[0], *tuple(symset)) != 0 else S.Zero
|
||
|
|
|
||
|
|
# make the replacement of x with x*t and see if t can be factored out
|
||
|
|
t = Dummy('t', positive=True) # It is sufficient that t > 0
|
||
|
|
eqs = separatevars(eq.subs([(i, t*i) for i in symset]), [t], dict=True)[t]
|
||
|
|
if eqs is S.One:
|
||
|
|
return S.Zero # there was no term with only t
|
||
|
|
i, d = eqs.as_independent(t, as_Add=False)
|
||
|
|
b, e = d.as_base_exp()
|
||
|
|
if b == t:
|
||
|
|
return e
|
||
|
|
|
||
|
|
|
||
|
|
def ode_2nd_power_series_ordinary(eq, func, order, match):
|
||
|
|
r"""
|
||
|
|
Gives a power series solution to a second order homogeneous differential
|
||
|
|
equation with polynomial coefficients at an ordinary point. A homogeneous
|
||
|
|
differential equation is of the form
|
||
|
|
|
||
|
|
.. math :: P(x)\frac{d^2y}{dx^2} + Q(x)\frac{dy}{dx} + R(x) y(x) = 0
|
||
|
|
|
||
|
|
For simplicity it is assumed that `P(x)`, `Q(x)` and `R(x)` are polynomials,
|
||
|
|
it is sufficient that `\frac{Q(x)}{P(x)}` and `\frac{R(x)}{P(x)}` exists at
|
||
|
|
`x_{0}`. A recurrence relation is obtained by substituting `y` as `\sum_{n=0}^\infty a_{n}x^{n}`,
|
||
|
|
in the differential equation, and equating the nth term. Using this relation
|
||
|
|
various terms can be generated.
|
||
|
|
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
>>> from sympy import dsolve, Function, pprint
|
||
|
|
>>> from sympy.abc import x
|
||
|
|
>>> f = Function("f")
|
||
|
|
>>> eq = f(x).diff(x, 2) + f(x)
|
||
|
|
>>> pprint(dsolve(eq, hint='2nd_power_series_ordinary'))
|
||
|
|
/ 4 2 \ / 2\
|
||
|
|
|x x | | x | / 6\
|
||
|
|
f(x) = C2*|-- - -- + 1| + C1*x*|1 - --| + O\x /
|
||
|
|
\24 2 / \ 6 /
|
||
|
|
|
||
|
|
|
||
|
|
References
|
||
|
|
==========
|
||
|
|
- https://tutorial.math.lamar.edu/Classes/DE/SeriesSolutions.aspx
|
||
|
|
- George E. Simmons, "Differential Equations with Applications and
|
||
|
|
Historical Notes", p.p 176 - 184
|
||
|
|
|
||
|
|
"""
|
||
|
|
x = func.args[0]
|
||
|
|
f = func.func
|
||
|
|
C0, C1 = get_numbered_constants(eq, num=2)
|
||
|
|
n = Dummy("n", integer=True)
|
||
|
|
s = Wild("s")
|
||
|
|
k = Wild("k", exclude=[x])
|
||
|
|
x0 = match['x0']
|
||
|
|
terms = match['terms']
|
||
|
|
p = match[match['a3']]
|
||
|
|
q = match[match['b3']]
|
||
|
|
r = match[match['c3']]
|
||
|
|
seriesdict = {}
|
||
|
|
recurr = Function("r")
|
||
|
|
|
||
|
|
# Generating the recurrence relation which works this way:
|
||
|
|
# for the second order term the summation begins at n = 2. The coefficients
|
||
|
|
# p is multiplied with an*(n - 1)*(n - 2)*x**n-2 and a substitution is made such that
|
||
|
|
# the exponent of x becomes n.
|
||
|
|
# For example, if p is x, then the second degree recurrence term is
|
||
|
|
# an*(n - 1)*(n - 2)*x**n-1, substituting (n - 1) as n, it transforms to
|
||
|
|
# an+1*n*(n - 1)*x**n.
|
||
|
|
# A similar process is done with the first order and zeroth order term.
|
||
|
|
|
||
|
|
coefflist = [(recurr(n), r), (n*recurr(n), q), (n*(n - 1)*recurr(n), p)]
|
||
|
|
for index, coeff in enumerate(coefflist):
|
||
|
|
if coeff[1]:
|
||
|
|
f2 = powsimp(expand((coeff[1]*(x - x0)**(n - index)).subs(x, x + x0)))
|
||
|
|
if f2.is_Add:
|
||
|
|
addargs = f2.args
|
||
|
|
else:
|
||
|
|
addargs = [f2]
|
||
|
|
for arg in addargs:
|
||
|
|
powm = arg.match(s*x**k)
|
||
|
|
term = coeff[0]*powm[s]
|
||
|
|
if not powm[k].is_Symbol:
|
||
|
|
term = term.subs(n, n - powm[k].as_independent(n)[0])
|
||
|
|
startind = powm[k].subs(n, index)
|
||
|
|
# Seeing if the startterm can be reduced further.
|
||
|
|
# If it vanishes for n lesser than startind, it is
|
||
|
|
# equal to summation from n.
|
||
|
|
if startind:
|
||
|
|
for i in reversed(range(startind)):
|
||
|
|
if not term.subs(n, i):
|
||
|
|
seriesdict[term] = i
|
||
|
|
else:
|
||
|
|
seriesdict[term] = i + 1
|
||
|
|
break
|
||
|
|
else:
|
||
|
|
seriesdict[term] = S.Zero
|
||
|
|
|
||
|
|
# Stripping of terms so that the sum starts with the same number.
|
||
|
|
teq = S.Zero
|
||
|
|
suminit = seriesdict.values()
|
||
|
|
rkeys = seriesdict.keys()
|
||
|
|
req = Add(*rkeys)
|
||
|
|
if any(suminit):
|
||
|
|
maxval = max(suminit)
|
||
|
|
for term in seriesdict:
|
||
|
|
val = seriesdict[term]
|
||
|
|
if val != maxval:
|
||
|
|
for i in range(val, maxval):
|
||
|
|
teq += term.subs(n, val)
|
||
|
|
|
||
|
|
finaldict = {}
|
||
|
|
if teq:
|
||
|
|
fargs = teq.atoms(AppliedUndef)
|
||
|
|
if len(fargs) == 1:
|
||
|
|
finaldict[fargs.pop()] = 0
|
||
|
|
else:
|
||
|
|
maxf = max(fargs, key = lambda x: x.args[0])
|
||
|
|
sol = solve(teq, maxf)
|
||
|
|
if isinstance(sol, list):
|
||
|
|
sol = sol[0]
|
||
|
|
finaldict[maxf] = sol
|
||
|
|
|
||
|
|
# Finding the recurrence relation in terms of the largest term.
|
||
|
|
fargs = req.atoms(AppliedUndef)
|
||
|
|
maxf = max(fargs, key = lambda x: x.args[0])
|
||
|
|
minf = min(fargs, key = lambda x: x.args[0])
|
||
|
|
if minf.args[0].is_Symbol:
|
||
|
|
startiter = 0
|
||
|
|
else:
|
||
|
|
startiter = -minf.args[0].as_independent(n)[0]
|
||
|
|
lhs = maxf
|
||
|
|
rhs = solve(req, maxf)
|
||
|
|
if isinstance(rhs, list):
|
||
|
|
rhs = rhs[0]
|
||
|
|
|
||
|
|
# Checking how many values are already present
|
||
|
|
tcounter = len([t for t in finaldict.values() if t])
|
||
|
|
|
||
|
|
for _ in range(tcounter, terms - 3): # Assuming c0 and c1 to be arbitrary
|
||
|
|
check = rhs.subs(n, startiter)
|
||
|
|
nlhs = lhs.subs(n, startiter)
|
||
|
|
nrhs = check.subs(finaldict)
|
||
|
|
finaldict[nlhs] = nrhs
|
||
|
|
startiter += 1
|
||
|
|
|
||
|
|
# Post processing
|
||
|
|
series = C0 + C1*(x - x0)
|
||
|
|
for term in finaldict:
|
||
|
|
if finaldict[term]:
|
||
|
|
fact = term.args[0]
|
||
|
|
series += (finaldict[term].subs([(recurr(0), C0), (recurr(1), C1)])*(
|
||
|
|
x - x0)**fact)
|
||
|
|
series = collect(expand_mul(series), [C0, C1]) + Order(x**terms)
|
||
|
|
return Eq(f(x), series)
|
||
|
|
|
||
|
|
|
||
|
|
def ode_2nd_power_series_regular(eq, func, order, match):
|
||
|
|
r"""
|
||
|
|
Gives a power series solution to a second order homogeneous differential
|
||
|
|
equation with polynomial coefficients at a regular point. A second order
|
||
|
|
homogeneous differential equation is of the form
|
||
|
|
|
||
|
|
.. math :: P(x)\frac{d^2y}{dx^2} + Q(x)\frac{dy}{dx} + R(x) y(x) = 0
|
||
|
|
|
||
|
|
A point is said to regular singular at `x0` if `x - x0\frac{Q(x)}{P(x)}`
|
||
|
|
and `(x - x0)^{2}\frac{R(x)}{P(x)}` are analytic at `x0`. For simplicity
|
||
|
|
`P(x)`, `Q(x)` and `R(x)` are assumed to be polynomials. The algorithm for
|
||
|
|
finding the power series solutions is:
|
||
|
|
|
||
|
|
1. Try expressing `(x - x0)P(x)` and `((x - x0)^{2})Q(x)` as power series
|
||
|
|
solutions about x0. Find `p0` and `q0` which are the constants of the
|
||
|
|
power series expansions.
|
||
|
|
2. Solve the indicial equation `f(m) = m(m - 1) + m*p0 + q0`, to obtain the
|
||
|
|
roots `m1` and `m2` of the indicial equation.
|
||
|
|
3. If `m1 - m2` is a non integer there exists two series solutions. If
|
||
|
|
`m1 = m2`, there exists only one solution. If `m1 - m2` is an integer,
|
||
|
|
then the existence of one solution is confirmed. The other solution may
|
||
|
|
or may not exist.
|
||
|
|
|
||
|
|
The power series solution is of the form `x^{m}\sum_{n=0}^\infty a_{n}x^{n}`. The
|
||
|
|
coefficients are determined by the following recurrence relation.
|
||
|
|
`a_{n} = -\frac{\sum_{k=0}^{n-1} q_{n-k} + (m + k)p_{n-k}}{f(m + n)}`. For the case
|
||
|
|
in which `m1 - m2` is an integer, it can be seen from the recurrence relation
|
||
|
|
that for the lower root `m`, when `n` equals the difference of both the
|
||
|
|
roots, the denominator becomes zero. So if the numerator is not equal to zero,
|
||
|
|
a second series solution exists.
|
||
|
|
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
>>> from sympy import dsolve, Function, pprint
|
||
|
|
>>> from sympy.abc import x
|
||
|
|
>>> f = Function("f")
|
||
|
|
>>> eq = x*(f(x).diff(x, 2)) + 2*(f(x).diff(x)) + x*f(x)
|
||
|
|
>>> pprint(dsolve(eq, hint='2nd_power_series_regular'))
|
||
|
|
/ 6 4 2 \
|
||
|
|
| x x x |
|
||
|
|
/ 4 2 \ C1*|- --- + -- - -- + 1|
|
||
|
|
| x x | \ 720 24 2 / / 6\
|
||
|
|
f(x) = C2*|--- - -- + 1| + ------------------------ + O\x /
|
||
|
|
\120 6 / x
|
||
|
|
|
||
|
|
|
||
|
|
References
|
||
|
|
==========
|
||
|
|
- George E. Simmons, "Differential Equations with Applications and
|
||
|
|
Historical Notes", p.p 176 - 184
|
||
|
|
|
||
|
|
"""
|
||
|
|
x = func.args[0]
|
||
|
|
f = func.func
|
||
|
|
C0, C1 = get_numbered_constants(eq, num=2)
|
||
|
|
m = Dummy("m") # for solving the indicial equation
|
||
|
|
x0 = match['x0']
|
||
|
|
terms = match['terms']
|
||
|
|
p = match['p']
|
||
|
|
q = match['q']
|
||
|
|
|
||
|
|
# Generating the indicial equation
|
||
|
|
indicial = []
|
||
|
|
for term in [p, q]:
|
||
|
|
if not term.has(x):
|
||
|
|
indicial.append(term)
|
||
|
|
else:
|
||
|
|
term = series(term, x=x, n=1, x0=x0)
|
||
|
|
if isinstance(term, Order):
|
||
|
|
indicial.append(S.Zero)
|
||
|
|
else:
|
||
|
|
for arg in term.args:
|
||
|
|
if not arg.has(x):
|
||
|
|
indicial.append(arg)
|
||
|
|
break
|
||
|
|
|
||
|
|
p0, q0 = indicial
|
||
|
|
sollist = solve(m*(m - 1) + m*p0 + q0, m)
|
||
|
|
if sollist and isinstance(sollist, list) and all(
|
||
|
|
sol.is_real for sol in sollist):
|
||
|
|
serdict1 = {}
|
||
|
|
serdict2 = {}
|
||
|
|
if len(sollist) == 1:
|
||
|
|
# Only one series solution exists in this case.
|
||
|
|
m1 = m2 = sollist.pop()
|
||
|
|
if terms-m1-1 <= 0:
|
||
|
|
return Eq(f(x), Order(terms))
|
||
|
|
serdict1 = _frobenius(terms-m1-1, m1, p0, q0, p, q, x0, x, C0)
|
||
|
|
|
||
|
|
else:
|
||
|
|
m1 = sollist[0]
|
||
|
|
m2 = sollist[1]
|
||
|
|
if m1 < m2:
|
||
|
|
m1, m2 = m2, m1
|
||
|
|
# Irrespective of whether m1 - m2 is an integer or not, one
|
||
|
|
# Frobenius series solution exists.
|
||
|
|
serdict1 = _frobenius(terms-m1-1, m1, p0, q0, p, q, x0, x, C0)
|
||
|
|
if not (m1 - m2).is_integer:
|
||
|
|
# Second frobenius series solution exists.
|
||
|
|
serdict2 = _frobenius(terms-m2-1, m2, p0, q0, p, q, x0, x, C1)
|
||
|
|
else:
|
||
|
|
# Check if second frobenius series solution exists.
|
||
|
|
serdict2 = _frobenius(terms-m2-1, m2, p0, q0, p, q, x0, x, C1, check=m1)
|
||
|
|
|
||
|
|
if serdict1:
|
||
|
|
finalseries1 = C0
|
||
|
|
for key in serdict1:
|
||
|
|
power = int(key.name[1:])
|
||
|
|
finalseries1 += serdict1[key]*(x - x0)**power
|
||
|
|
finalseries1 = (x - x0)**m1*finalseries1
|
||
|
|
finalseries2 = S.Zero
|
||
|
|
if serdict2:
|
||
|
|
for key in serdict2:
|
||
|
|
power = int(key.name[1:])
|
||
|
|
finalseries2 += serdict2[key]*(x - x0)**power
|
||
|
|
finalseries2 += C1
|
||
|
|
finalseries2 = (x - x0)**m2*finalseries2
|
||
|
|
return Eq(f(x), collect(finalseries1 + finalseries2,
|
||
|
|
[C0, C1]) + Order(x**terms))
|
||
|
|
|
||
|
|
|
||
|
|
def _frobenius(n, m, p0, q0, p, q, x0, x, c, check=None):
|
||
|
|
r"""
|
||
|
|
Returns a dict with keys as coefficients and values as their values in terms of C0
|
||
|
|
"""
|
||
|
|
n = int(n)
|
||
|
|
# In cases where m1 - m2 is not an integer
|
||
|
|
m2 = check
|
||
|
|
|
||
|
|
d = Dummy("d")
|
||
|
|
numsyms = numbered_symbols("C", start=0)
|
||
|
|
numsyms = [next(numsyms) for i in range(n + 1)]
|
||
|
|
serlist = []
|
||
|
|
for ser in [p, q]:
|
||
|
|
# Order term not present
|
||
|
|
if ser.is_polynomial(x) and Poly(ser, x).degree() <= n:
|
||
|
|
if x0:
|
||
|
|
ser = ser.subs(x, x + x0)
|
||
|
|
dict_ = Poly(ser, x).as_dict()
|
||
|
|
# Order term present
|
||
|
|
else:
|
||
|
|
tseries = series(ser, x=x0, n=n+1)
|
||
|
|
# Removing order
|
||
|
|
dict_ = Poly(list(ordered(tseries.args))[: -1], x).as_dict()
|
||
|
|
# Fill in with zeros, if coefficients are zero.
|
||
|
|
for i in range(n + 1):
|
||
|
|
if (i,) not in dict_:
|
||
|
|
dict_[(i,)] = S.Zero
|
||
|
|
serlist.append(dict_)
|
||
|
|
|
||
|
|
pseries = serlist[0]
|
||
|
|
qseries = serlist[1]
|
||
|
|
indicial = d*(d - 1) + d*p0 + q0
|
||
|
|
frobdict = {}
|
||
|
|
for i in range(1, n + 1):
|
||
|
|
num = c*(m*pseries[(i,)] + qseries[(i,)])
|
||
|
|
for j in range(1, i):
|
||
|
|
sym = Symbol("C" + str(j))
|
||
|
|
num += frobdict[sym]*((m + j)*pseries[(i - j,)] + qseries[(i - j,)])
|
||
|
|
|
||
|
|
# Checking for cases when m1 - m2 is an integer. If num equals zero
|
||
|
|
# then a second Frobenius series solution cannot be found. If num is not zero
|
||
|
|
# then set constant as zero and proceed.
|
||
|
|
if m2 is not None and i == m2 - m:
|
||
|
|
if num:
|
||
|
|
return False
|
||
|
|
else:
|
||
|
|
frobdict[numsyms[i]] = S.Zero
|
||
|
|
else:
|
||
|
|
frobdict[numsyms[i]] = -num/(indicial.subs(d, m+i))
|
||
|
|
|
||
|
|
return frobdict
|
||
|
|
|
||
|
|
def _remove_redundant_solutions(eq, solns, order, var):
|
||
|
|
r"""
|
||
|
|
Remove redundant solutions from the set of solutions.
|
||
|
|
|
||
|
|
This function is needed because otherwise dsolve can return
|
||
|
|
redundant solutions. As an example consider:
|
||
|
|
|
||
|
|
eq = Eq((f(x).diff(x, 2))*f(x).diff(x), 0)
|
||
|
|
|
||
|
|
There are two ways to find solutions to eq. The first is to solve f(x).diff(x, 2) = 0
|
||
|
|
leading to solution f(x)=C1 + C2*x. The second is to solve the equation f(x).diff(x) = 0
|
||
|
|
leading to the solution f(x) = C1. In this particular case we then see
|
||
|
|
that the second solution is a special case of the first and we do not
|
||
|
|
want to return it.
|
||
|
|
|
||
|
|
This does not always happen. If we have
|
||
|
|
|
||
|
|
eq = Eq((f(x)**2-4)*(f(x).diff(x)-4), 0)
|
||
|
|
|
||
|
|
then we get the algebraic solution f(x) = [-2, 2] and the integral solution
|
||
|
|
f(x) = x + C1 and in this case the two solutions are not equivalent wrt
|
||
|
|
initial conditions so both should be returned.
|
||
|
|
"""
|
||
|
|
def is_special_case_of(soln1, soln2):
|
||
|
|
return _is_special_case_of(soln1, soln2, eq, order, var)
|
||
|
|
|
||
|
|
unique_solns = []
|
||
|
|
for soln1 in solns:
|
||
|
|
for soln2 in unique_solns[:]:
|
||
|
|
if is_special_case_of(soln1, soln2):
|
||
|
|
break
|
||
|
|
elif is_special_case_of(soln2, soln1):
|
||
|
|
unique_solns.remove(soln2)
|
||
|
|
else:
|
||
|
|
unique_solns.append(soln1)
|
||
|
|
|
||
|
|
return unique_solns
|
||
|
|
|
||
|
|
def _is_special_case_of(soln1, soln2, eq, order, var):
|
||
|
|
r"""
|
||
|
|
True if soln1 is found to be a special case of soln2 wrt some value of the
|
||
|
|
constants that appear in soln2. False otherwise.
|
||
|
|
"""
|
||
|
|
# The solutions returned by dsolve may be given explicitly or implicitly.
|
||
|
|
# We will equate the sol1=(soln1.rhs - soln1.lhs), sol2=(soln2.rhs - soln2.lhs)
|
||
|
|
# of the two solutions.
|
||
|
|
#
|
||
|
|
# Since this is supposed to hold for all x it also holds for derivatives.
|
||
|
|
# For an order n ode we should be able to differentiate
|
||
|
|
# each solution n times to get n+1 equations.
|
||
|
|
#
|
||
|
|
# We then try to solve those n+1 equations for the integrations constants
|
||
|
|
# in sol2. If we can find a solution that does not depend on x then it
|
||
|
|
# means that some value of the constants in sol1 is a special case of
|
||
|
|
# sol2 corresponding to a particular choice of the integration constants.
|
||
|
|
|
||
|
|
# In case the solution is in implicit form we subtract the sides
|
||
|
|
soln1 = soln1.rhs - soln1.lhs
|
||
|
|
soln2 = soln2.rhs - soln2.lhs
|
||
|
|
|
||
|
|
# Work for the series solution
|
||
|
|
if soln1.has(Order) and soln2.has(Order):
|
||
|
|
if soln1.getO() == soln2.getO():
|
||
|
|
soln1 = soln1.removeO()
|
||
|
|
soln2 = soln2.removeO()
|
||
|
|
else:
|
||
|
|
return False
|
||
|
|
elif soln1.has(Order) or soln2.has(Order):
|
||
|
|
return False
|
||
|
|
|
||
|
|
constants1 = soln1.free_symbols.difference(eq.free_symbols)
|
||
|
|
constants2 = soln2.free_symbols.difference(eq.free_symbols)
|
||
|
|
|
||
|
|
constants1_new = get_numbered_constants(Tuple(soln1, soln2), len(constants1))
|
||
|
|
if len(constants1) == 1:
|
||
|
|
constants1_new = {constants1_new}
|
||
|
|
for c_old, c_new in zip(constants1, constants1_new):
|
||
|
|
soln1 = soln1.subs(c_old, c_new)
|
||
|
|
|
||
|
|
# n equations for sol1 = sol2, sol1'=sol2', ...
|
||
|
|
lhs = soln1
|
||
|
|
rhs = soln2
|
||
|
|
eqns = [Eq(lhs, rhs)]
|
||
|
|
for n in range(1, order):
|
||
|
|
lhs = lhs.diff(var)
|
||
|
|
rhs = rhs.diff(var)
|
||
|
|
eq = Eq(lhs, rhs)
|
||
|
|
eqns.append(eq)
|
||
|
|
|
||
|
|
# BooleanTrue/False awkwardly show up for trivial equations
|
||
|
|
if any(isinstance(eq, BooleanFalse) for eq in eqns):
|
||
|
|
return False
|
||
|
|
eqns = [eq for eq in eqns if not isinstance(eq, BooleanTrue)]
|
||
|
|
|
||
|
|
try:
|
||
|
|
constant_solns = solve(eqns, constants2)
|
||
|
|
except NotImplementedError:
|
||
|
|
return False
|
||
|
|
|
||
|
|
# Sometimes returns a dict and sometimes a list of dicts
|
||
|
|
if isinstance(constant_solns, dict):
|
||
|
|
constant_solns = [constant_solns]
|
||
|
|
|
||
|
|
# after solving the issue 17418, maybe we don't need the following checksol code.
|
||
|
|
for constant_soln in constant_solns:
|
||
|
|
for eq in eqns:
|
||
|
|
eq=eq.rhs-eq.lhs
|
||
|
|
if checksol(eq, constant_soln) is not True:
|
||
|
|
return False
|
||
|
|
|
||
|
|
# If any solution gives all constants as expressions that don't depend on
|
||
|
|
# x then there exists constants for soln2 that give soln1
|
||
|
|
for constant_soln in constant_solns:
|
||
|
|
if not any(c.has(var) for c in constant_soln.values()):
|
||
|
|
return True
|
||
|
|
|
||
|
|
return False
|
||
|
|
|
||
|
|
|
||
|
|
def ode_1st_power_series(eq, func, order, match):
|
||
|
|
r"""
|
||
|
|
The power series solution is a method which gives the Taylor series expansion
|
||
|
|
to the solution of a differential equation.
|
||
|
|
|
||
|
|
For a first order differential equation `\frac{dy}{dx} = h(x, y)`, a power
|
||
|
|
series solution exists at a point `x = x_{0}` if `h(x, y)` is analytic at `x_{0}`.
|
||
|
|
The solution is given by
|
||
|
|
|
||
|
|
.. math:: y(x) = y(x_{0}) + \sum_{n = 1}^{\infty} \frac{F_{n}(x_{0},b)(x - x_{0})^n}{n!},
|
||
|
|
|
||
|
|
where `y(x_{0}) = b` is the value of y at the initial value of `x_{0}`.
|
||
|
|
To compute the values of the `F_{n}(x_{0},b)` the following algorithm is
|
||
|
|
followed, until the required number of terms are generated.
|
||
|
|
|
||
|
|
1. `F_1 = h(x_{0}, b)`
|
||
|
|
2. `F_{n+1} = \frac{\partial F_{n}}{\partial x} + \frac{\partial F_{n}}{\partial y}F_{1}`
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
>>> from sympy import Function, pprint, exp, dsolve
|
||
|
|
>>> from sympy.abc import x
|
||
|
|
>>> f = Function('f')
|
||
|
|
>>> eq = exp(x)*(f(x).diff(x)) - f(x)
|
||
|
|
>>> pprint(dsolve(eq, hint='1st_power_series'))
|
||
|
|
3 4 5
|
||
|
|
C1*x C1*x C1*x / 6\
|
||
|
|
f(x) = C1 + C1*x - ----- + ----- + ----- + O\x /
|
||
|
|
6 24 60
|
||
|
|
|
||
|
|
|
||
|
|
References
|
||
|
|
==========
|
||
|
|
|
||
|
|
- Travis W. Walker, Analytic power series technique for solving first-order
|
||
|
|
differential equations, p.p 17, 18
|
||
|
|
|
||
|
|
"""
|
||
|
|
x = func.args[0]
|
||
|
|
y = match['y']
|
||
|
|
f = func.func
|
||
|
|
h = -match[match['d']]/match[match['e']]
|
||
|
|
point = match['f0']
|
||
|
|
value = match['f0val']
|
||
|
|
terms = match['terms']
|
||
|
|
|
||
|
|
# First term
|
||
|
|
F = h
|
||
|
|
if not h:
|
||
|
|
return Eq(f(x), value)
|
||
|
|
|
||
|
|
# Initialization
|
||
|
|
series = value
|
||
|
|
if terms > 1:
|
||
|
|
hc = h.subs({x: point, y: value})
|
||
|
|
if hc.has(oo) or hc.has(nan) or hc.has(zoo):
|
||
|
|
# Derivative does not exist, not analytic
|
||
|
|
return Eq(f(x), oo)
|
||
|
|
elif hc:
|
||
|
|
series += hc*(x - point)
|
||
|
|
|
||
|
|
for factcount in range(2, terms):
|
||
|
|
Fnew = F.diff(x) + F.diff(y)*h
|
||
|
|
Fnewc = Fnew.subs({x: point, y: value})
|
||
|
|
# Same logic as above
|
||
|
|
if Fnewc.has(oo) or Fnewc.has(nan) or Fnewc.has(-oo) or Fnewc.has(zoo):
|
||
|
|
return Eq(f(x), oo)
|
||
|
|
series += Fnewc*((x - point)**factcount)/factorial(factcount)
|
||
|
|
F = Fnew
|
||
|
|
series += Order(x**terms)
|
||
|
|
return Eq(f(x), series)
|
||
|
|
|
||
|
|
|
||
|
|
def checkinfsol(eq, infinitesimals, func=None, order=None):
|
||
|
|
r"""
|
||
|
|
This function is used to check if the given infinitesimals are the
|
||
|
|
actual infinitesimals of the given first order differential equation.
|
||
|
|
This method is specific to the Lie Group Solver of ODEs.
|
||
|
|
|
||
|
|
As of now, it simply checks, by substituting the infinitesimals in the
|
||
|
|
partial differential equation.
|
||
|
|
|
||
|
|
|
||
|
|
.. math:: \frac{\partial \eta}{\partial x} + \left(\frac{\partial \eta}{\partial y}
|
||
|
|
- \frac{\partial \xi}{\partial x}\right)*h
|
||
|
|
- \frac{\partial \xi}{\partial y}*h^{2}
|
||
|
|
- \xi\frac{\partial h}{\partial x} - \eta\frac{\partial h}{\partial y} = 0
|
||
|
|
|
||
|
|
|
||
|
|
where `\eta`, and `\xi` are the infinitesimals and `h(x,y) = \frac{dy}{dx}`
|
||
|
|
|
||
|
|
The infinitesimals should be given in the form of a list of dicts
|
||
|
|
``[{xi(x, y): inf, eta(x, y): inf}]``, corresponding to the
|
||
|
|
output of the function infinitesimals. It returns a list
|
||
|
|
of values of the form ``[(True/False, sol)]`` where ``sol`` is the value
|
||
|
|
obtained after substituting the infinitesimals in the PDE. If it
|
||
|
|
is ``True``, then ``sol`` would be 0.
|
||
|
|
|
||
|
|
"""
|
||
|
|
if isinstance(eq, Equality):
|
||
|
|
eq = eq.lhs - eq.rhs
|
||
|
|
if not func:
|
||
|
|
eq, func = _preprocess(eq)
|
||
|
|
variables = func.args
|
||
|
|
if len(variables) != 1:
|
||
|
|
raise ValueError("ODE's have only one independent variable")
|
||
|
|
else:
|
||
|
|
x = variables[0]
|
||
|
|
if not order:
|
||
|
|
order = ode_order(eq, func)
|
||
|
|
if order != 1:
|
||
|
|
raise NotImplementedError("Lie groups solver has been implemented "
|
||
|
|
"only for first order differential equations")
|
||
|
|
else:
|
||
|
|
df = func.diff(x)
|
||
|
|
a = Wild('a', exclude = [df])
|
||
|
|
b = Wild('b', exclude = [df])
|
||
|
|
match = collect(expand(eq), df).match(a*df + b)
|
||
|
|
|
||
|
|
if match:
|
||
|
|
h = -simplify(match[b]/match[a])
|
||
|
|
else:
|
||
|
|
try:
|
||
|
|
sol = solve(eq, df)
|
||
|
|
except NotImplementedError:
|
||
|
|
raise NotImplementedError("Infinitesimals for the "
|
||
|
|
"first order ODE could not be found")
|
||
|
|
else:
|
||
|
|
h = sol[0] # Find infinitesimals for one solution
|
||
|
|
|
||
|
|
y = Dummy('y')
|
||
|
|
h = h.subs(func, y)
|
||
|
|
xi = Function('xi')(x, y)
|
||
|
|
eta = Function('eta')(x, y)
|
||
|
|
dxi = Function('xi')(x, func)
|
||
|
|
deta = Function('eta')(x, func)
|
||
|
|
pde = (eta.diff(x) + (eta.diff(y) - xi.diff(x))*h -
|
||
|
|
(xi.diff(y))*h**2 - xi*(h.diff(x)) - eta*(h.diff(y)))
|
||
|
|
soltup = []
|
||
|
|
for sol in infinitesimals:
|
||
|
|
tsol = {xi: S(sol[dxi]).subs(func, y),
|
||
|
|
eta: S(sol[deta]).subs(func, y)}
|
||
|
|
sol = simplify(pde.subs(tsol).doit())
|
||
|
|
if sol:
|
||
|
|
soltup.append((False, sol.subs(y, func)))
|
||
|
|
else:
|
||
|
|
soltup.append((True, 0))
|
||
|
|
return soltup
|
||
|
|
|
||
|
|
|
||
|
|
def sysode_linear_2eq_order1(match_):
|
||
|
|
x = match_['func'][0].func
|
||
|
|
y = match_['func'][1].func
|
||
|
|
func = match_['func']
|
||
|
|
fc = match_['func_coeff']
|
||
|
|
eq = match_['eq']
|
||
|
|
r = {}
|
||
|
|
t = list(list(eq[0].atoms(Derivative))[0].atoms(Symbol))[0]
|
||
|
|
for i in range(2):
|
||
|
|
eq[i] = Add(*[terms/fc[i,func[i],1] for terms in Add.make_args(eq[i])])
|
||
|
|
|
||
|
|
# for equations Eq(a1*diff(x(t),t), a*x(t) + b*y(t) + k1)
|
||
|
|
# and Eq(a2*diff(x(t),t), c*x(t) + d*y(t) + k2)
|
||
|
|
r['a'] = -fc[0,x(t),0]/fc[0,x(t),1]
|
||
|
|
r['c'] = -fc[1,x(t),0]/fc[1,y(t),1]
|
||
|
|
r['b'] = -fc[0,y(t),0]/fc[0,x(t),1]
|
||
|
|
r['d'] = -fc[1,y(t),0]/fc[1,y(t),1]
|
||
|
|
forcing = [S.Zero,S.Zero]
|
||
|
|
for i in range(2):
|
||
|
|
for j in Add.make_args(eq[i]):
|
||
|
|
if not j.has(x(t), y(t)):
|
||
|
|
forcing[i] += j
|
||
|
|
if not (forcing[0].has(t) or forcing[1].has(t)):
|
||
|
|
r['k1'] = forcing[0]
|
||
|
|
r['k2'] = forcing[1]
|
||
|
|
else:
|
||
|
|
raise NotImplementedError("Only homogeneous problems are supported" +
|
||
|
|
" (and constant inhomogeneity)")
|
||
|
|
|
||
|
|
if match_['type_of_equation'] == 'type6':
|
||
|
|
sol = _linear_2eq_order1_type6(x, y, t, r, eq)
|
||
|
|
if match_['type_of_equation'] == 'type7':
|
||
|
|
sol = _linear_2eq_order1_type7(x, y, t, r, eq)
|
||
|
|
return sol
|
||
|
|
|
||
|
|
def _linear_2eq_order1_type6(x, y, t, r, eq):
|
||
|
|
r"""
|
||
|
|
The equations of this type of ode are .
|
||
|
|
|
||
|
|
.. math:: x' = f(t) x + g(t) y
|
||
|
|
|
||
|
|
.. math:: y' = a [f(t) + a h(t)] x + a [g(t) - h(t)] y
|
||
|
|
|
||
|
|
This is solved by first multiplying the first equation by `-a` and adding
|
||
|
|
it to the second equation to obtain
|
||
|
|
|
||
|
|
.. math:: y' - a x' = -a h(t) (y - a x)
|
||
|
|
|
||
|
|
Setting `U = y - ax` and integrating the equation we arrive at
|
||
|
|
|
||
|
|
.. math:: y - ax = C_1 e^{-a \int h(t) \,dt}
|
||
|
|
|
||
|
|
and on substituting the value of y in first equation give rise to first order ODEs. After solving for
|
||
|
|
`x`, we can obtain `y` by substituting the value of `x` in second equation.
|
||
|
|
|
||
|
|
"""
|
||
|
|
C1, C2, C3, C4 = get_numbered_constants(eq, num=4)
|
||
|
|
p = 0
|
||
|
|
q = 0
|
||
|
|
p1 = cancel(r['c']/cancel(r['c']/r['d']).as_numer_denom()[0])
|
||
|
|
p2 = cancel(r['a']/cancel(r['a']/r['b']).as_numer_denom()[0])
|
||
|
|
for n, i in enumerate([p1, p2]):
|
||
|
|
for j in Mul.make_args(collect_const(i)):
|
||
|
|
if not j.has(t):
|
||
|
|
q = j
|
||
|
|
if q!=0 and n==0:
|
||
|
|
if ((r['c']/j - r['a'])/(r['b'] - r['d']/j)) == j:
|
||
|
|
p = 1
|
||
|
|
s = j
|
||
|
|
break
|
||
|
|
if q!=0 and n==1:
|
||
|
|
if ((r['a']/j - r['c'])/(r['d'] - r['b']/j)) == j:
|
||
|
|
p = 2
|
||
|
|
s = j
|
||
|
|
break
|
||
|
|
|
||
|
|
if p == 1:
|
||
|
|
equ = diff(x(t),t) - r['a']*x(t) - r['b']*(s*x(t) + C1*exp(-s*Integral(r['b'] - r['d']/s, t)))
|
||
|
|
hint1 = classify_ode(equ)[1]
|
||
|
|
sol1 = dsolve(equ, hint=hint1+'_Integral').rhs
|
||
|
|
sol2 = s*sol1 + C1*exp(-s*Integral(r['b'] - r['d']/s, t))
|
||
|
|
elif p ==2:
|
||
|
|
equ = diff(y(t),t) - r['c']*y(t) - r['d']*s*y(t) + C1*exp(-s*Integral(r['d'] - r['b']/s, t))
|
||
|
|
hint1 = classify_ode(equ)[1]
|
||
|
|
sol2 = dsolve(equ, hint=hint1+'_Integral').rhs
|
||
|
|
sol1 = s*sol2 + C1*exp(-s*Integral(r['d'] - r['b']/s, t))
|
||
|
|
return [Eq(x(t), sol1), Eq(y(t), sol2)]
|
||
|
|
|
||
|
|
def _linear_2eq_order1_type7(x, y, t, r, eq):
|
||
|
|
r"""
|
||
|
|
The equations of this type of ode are .
|
||
|
|
|
||
|
|
.. math:: x' = f(t) x + g(t) y
|
||
|
|
|
||
|
|
.. math:: y' = h(t) x + p(t) y
|
||
|
|
|
||
|
|
Differentiating the first equation and substituting the value of `y`
|
||
|
|
from second equation will give a second-order linear equation
|
||
|
|
|
||
|
|
.. math:: g x'' - (fg + gp + g') x' + (fgp - g^{2} h + f g' - f' g) x = 0
|
||
|
|
|
||
|
|
This above equation can be easily integrated if following conditions are satisfied.
|
||
|
|
|
||
|
|
1. `fgp - g^{2} h + f g' - f' g = 0`
|
||
|
|
|
||
|
|
2. `fgp - g^{2} h + f g' - f' g = ag, fg + gp + g' = bg`
|
||
|
|
|
||
|
|
If first condition is satisfied then it is solved by current dsolve solver and in second case it becomes
|
||
|
|
a constant coefficient differential equation which is also solved by current solver.
|
||
|
|
|
||
|
|
Otherwise if the above condition fails then,
|
||
|
|
a particular solution is assumed as `x = x_0(t)` and `y = y_0(t)`
|
||
|
|
Then the general solution is expressed as
|
||
|
|
|
||
|
|
.. math:: x = C_1 x_0(t) + C_2 x_0(t) \int \frac{g(t) F(t) P(t)}{x_0^{2}(t)} \,dt
|
||
|
|
|
||
|
|
.. math:: y = C_1 y_0(t) + C_2 [\frac{F(t) P(t)}{x_0(t)} + y_0(t) \int \frac{g(t) F(t) P(t)}{x_0^{2}(t)} \,dt]
|
||
|
|
|
||
|
|
where C1 and C2 are arbitrary constants and
|
||
|
|
|
||
|
|
.. math:: F(t) = e^{\int f(t) \,dt}, P(t) = e^{\int p(t) \,dt}
|
||
|
|
|
||
|
|
"""
|
||
|
|
C1, C2, C3, C4 = get_numbered_constants(eq, num=4)
|
||
|
|
e1 = r['a']*r['b']*r['c'] - r['b']**2*r['c'] + r['a']*diff(r['b'],t) - diff(r['a'],t)*r['b']
|
||
|
|
e2 = r['a']*r['c']*r['d'] - r['b']*r['c']**2 + diff(r['c'],t)*r['d'] - r['c']*diff(r['d'],t)
|
||
|
|
m1 = r['a']*r['b'] + r['b']*r['d'] + diff(r['b'],t)
|
||
|
|
m2 = r['a']*r['c'] + r['c']*r['d'] + diff(r['c'],t)
|
||
|
|
if e1 == 0:
|
||
|
|
sol1 = dsolve(r['b']*diff(x(t),t,t) - m1*diff(x(t),t)).rhs
|
||
|
|
sol2 = dsolve(diff(y(t),t) - r['c']*sol1 - r['d']*y(t)).rhs
|
||
|
|
elif e2 == 0:
|
||
|
|
sol2 = dsolve(r['c']*diff(y(t),t,t) - m2*diff(y(t),t)).rhs
|
||
|
|
sol1 = dsolve(diff(x(t),t) - r['a']*x(t) - r['b']*sol2).rhs
|
||
|
|
elif not (e1/r['b']).has(t) and not (m1/r['b']).has(t):
|
||
|
|
sol1 = dsolve(diff(x(t),t,t) - (m1/r['b'])*diff(x(t),t) - (e1/r['b'])*x(t)).rhs
|
||
|
|
sol2 = dsolve(diff(y(t),t) - r['c']*sol1 - r['d']*y(t)).rhs
|
||
|
|
elif not (e2/r['c']).has(t) and not (m2/r['c']).has(t):
|
||
|
|
sol2 = dsolve(diff(y(t),t,t) - (m2/r['c'])*diff(y(t),t) - (e2/r['c'])*y(t)).rhs
|
||
|
|
sol1 = dsolve(diff(x(t),t) - r['a']*x(t) - r['b']*sol2).rhs
|
||
|
|
else:
|
||
|
|
x0 = Function('x0')(t) # x0 and y0 being particular solutions
|
||
|
|
y0 = Function('y0')(t)
|
||
|
|
F = exp(Integral(r['a'],t))
|
||
|
|
P = exp(Integral(r['d'],t))
|
||
|
|
sol1 = C1*x0 + C2*x0*Integral(r['b']*F*P/x0**2, t)
|
||
|
|
sol2 = C1*y0 + C2*(F*P/x0 + y0*Integral(r['b']*F*P/x0**2, t))
|
||
|
|
return [Eq(x(t), sol1), Eq(y(t), sol2)]
|
||
|
|
|
||
|
|
|
||
|
|
def sysode_nonlinear_2eq_order1(match_):
|
||
|
|
func = match_['func']
|
||
|
|
eq = match_['eq']
|
||
|
|
fc = match_['func_coeff']
|
||
|
|
t = list(list(eq[0].atoms(Derivative))[0].atoms(Symbol))[0]
|
||
|
|
if match_['type_of_equation'] == 'type5':
|
||
|
|
sol = _nonlinear_2eq_order1_type5(func, t, eq)
|
||
|
|
return sol
|
||
|
|
x = func[0].func
|
||
|
|
y = func[1].func
|
||
|
|
for i in range(2):
|
||
|
|
eqs = 0
|
||
|
|
for terms in Add.make_args(eq[i]):
|
||
|
|
eqs += terms/fc[i,func[i],1]
|
||
|
|
eq[i] = eqs
|
||
|
|
if match_['type_of_equation'] == 'type1':
|
||
|
|
sol = _nonlinear_2eq_order1_type1(x, y, t, eq)
|
||
|
|
elif match_['type_of_equation'] == 'type2':
|
||
|
|
sol = _nonlinear_2eq_order1_type2(x, y, t, eq)
|
||
|
|
elif match_['type_of_equation'] == 'type3':
|
||
|
|
sol = _nonlinear_2eq_order1_type3(x, y, t, eq)
|
||
|
|
elif match_['type_of_equation'] == 'type4':
|
||
|
|
sol = _nonlinear_2eq_order1_type4(x, y, t, eq)
|
||
|
|
return sol
|
||
|
|
|
||
|
|
|
||
|
|
def _nonlinear_2eq_order1_type1(x, y, t, eq):
|
||
|
|
r"""
|
||
|
|
Equations:
|
||
|
|
|
||
|
|
.. math:: x' = x^n F(x,y)
|
||
|
|
|
||
|
|
.. math:: y' = g(y) F(x,y)
|
||
|
|
|
||
|
|
Solution:
|
||
|
|
|
||
|
|
.. math:: x = \varphi(y), \int \frac{1}{g(y) F(\varphi(y),y)} \,dy = t + C_2
|
||
|
|
|
||
|
|
where
|
||
|
|
|
||
|
|
if `n \neq 1`
|
||
|
|
|
||
|
|
.. math:: \varphi = [C_1 + (1-n) \int \frac{1}{g(y)} \,dy]^{\frac{1}{1-n}}
|
||
|
|
|
||
|
|
if `n = 1`
|
||
|
|
|
||
|
|
.. math:: \varphi = C_1 e^{\int \frac{1}{g(y)} \,dy}
|
||
|
|
|
||
|
|
where `C_1` and `C_2` are arbitrary constants.
|
||
|
|
|
||
|
|
"""
|
||
|
|
C1, C2 = get_numbered_constants(eq, num=2)
|
||
|
|
n = Wild('n', exclude=[x(t),y(t)])
|
||
|
|
f = Wild('f')
|
||
|
|
u, v = symbols('u, v')
|
||
|
|
r = eq[0].match(diff(x(t),t) - x(t)**n*f)
|
||
|
|
g = ((diff(y(t),t) - eq[1])/r[f]).subs(y(t),v)
|
||
|
|
F = r[f].subs(x(t),u).subs(y(t),v)
|
||
|
|
n = r[n]
|
||
|
|
if n!=1:
|
||
|
|
phi = (C1 + (1-n)*Integral(1/g, v))**(1/(1-n))
|
||
|
|
else:
|
||
|
|
phi = C1*exp(Integral(1/g, v))
|
||
|
|
phi = phi.doit()
|
||
|
|
sol2 = solve(Integral(1/(g*F.subs(u,phi)), v).doit() - t - C2, v)
|
||
|
|
sol = []
|
||
|
|
for sols in sol2:
|
||
|
|
sol.append(Eq(x(t),phi.subs(v, sols)))
|
||
|
|
sol.append(Eq(y(t), sols))
|
||
|
|
return sol
|
||
|
|
|
||
|
|
def _nonlinear_2eq_order1_type2(x, y, t, eq):
|
||
|
|
r"""
|
||
|
|
Equations:
|
||
|
|
|
||
|
|
.. math:: x' = e^{\lambda x} F(x,y)
|
||
|
|
|
||
|
|
.. math:: y' = g(y) F(x,y)
|
||
|
|
|
||
|
|
Solution:
|
||
|
|
|
||
|
|
.. math:: x = \varphi(y), \int \frac{1}{g(y) F(\varphi(y),y)} \,dy = t + C_2
|
||
|
|
|
||
|
|
where
|
||
|
|
|
||
|
|
if `\lambda \neq 0`
|
||
|
|
|
||
|
|
.. math:: \varphi = -\frac{1}{\lambda} log(C_1 - \lambda \int \frac{1}{g(y)} \,dy)
|
||
|
|
|
||
|
|
if `\lambda = 0`
|
||
|
|
|
||
|
|
.. math:: \varphi = C_1 + \int \frac{1}{g(y)} \,dy
|
||
|
|
|
||
|
|
where `C_1` and `C_2` are arbitrary constants.
|
||
|
|
|
||
|
|
"""
|
||
|
|
C1, C2 = get_numbered_constants(eq, num=2)
|
||
|
|
n = Wild('n', exclude=[x(t),y(t)])
|
||
|
|
f = Wild('f')
|
||
|
|
u, v = symbols('u, v')
|
||
|
|
r = eq[0].match(diff(x(t),t) - exp(n*x(t))*f)
|
||
|
|
g = ((diff(y(t),t) - eq[1])/r[f]).subs(y(t),v)
|
||
|
|
F = r[f].subs(x(t),u).subs(y(t),v)
|
||
|
|
n = r[n]
|
||
|
|
if n:
|
||
|
|
phi = -1/n*log(C1 - n*Integral(1/g, v))
|
||
|
|
else:
|
||
|
|
phi = C1 + Integral(1/g, v)
|
||
|
|
phi = phi.doit()
|
||
|
|
sol2 = solve(Integral(1/(g*F.subs(u,phi)), v).doit() - t - C2, v)
|
||
|
|
sol = []
|
||
|
|
for sols in sol2:
|
||
|
|
sol.append(Eq(x(t),phi.subs(v, sols)))
|
||
|
|
sol.append(Eq(y(t), sols))
|
||
|
|
return sol
|
||
|
|
|
||
|
|
def _nonlinear_2eq_order1_type3(x, y, t, eq):
|
||
|
|
r"""
|
||
|
|
Autonomous system of general form
|
||
|
|
|
||
|
|
.. math:: x' = F(x,y)
|
||
|
|
|
||
|
|
.. math:: y' = G(x,y)
|
||
|
|
|
||
|
|
Assuming `y = y(x, C_1)` where `C_1` is an arbitrary constant is the general
|
||
|
|
solution of the first-order equation
|
||
|
|
|
||
|
|
.. math:: F(x,y) y'_x = G(x,y)
|
||
|
|
|
||
|
|
Then the general solution of the original system of equations has the form
|
||
|
|
|
||
|
|
.. math:: \int \frac{1}{F(x,y(x,C_1))} \,dx = t + C_1
|
||
|
|
|
||
|
|
"""
|
||
|
|
C1, C2, C3, C4 = get_numbered_constants(eq, num=4)
|
||
|
|
v = Function('v')
|
||
|
|
u = Symbol('u')
|
||
|
|
f = Wild('f')
|
||
|
|
g = Wild('g')
|
||
|
|
r1 = eq[0].match(diff(x(t),t) - f)
|
||
|
|
r2 = eq[1].match(diff(y(t),t) - g)
|
||
|
|
F = r1[f].subs(x(t), u).subs(y(t), v(u))
|
||
|
|
G = r2[g].subs(x(t), u).subs(y(t), v(u))
|
||
|
|
sol2r = dsolve(Eq(diff(v(u), u), G/F))
|
||
|
|
if isinstance(sol2r, Equality):
|
||
|
|
sol2r = [sol2r]
|
||
|
|
for sol2s in sol2r:
|
||
|
|
sol1 = solve(Integral(1/F.subs(v(u), sol2s.rhs), u).doit() - t - C2, u)
|
||
|
|
sol = []
|
||
|
|
for sols in sol1:
|
||
|
|
sol.append(Eq(x(t), sols))
|
||
|
|
sol.append(Eq(y(t), (sol2s.rhs).subs(u, sols)))
|
||
|
|
return sol
|
||
|
|
|
||
|
|
def _nonlinear_2eq_order1_type4(x, y, t, eq):
|
||
|
|
r"""
|
||
|
|
Equation:
|
||
|
|
|
||
|
|
.. math:: x' = f_1(x) g_1(y) \phi(x,y,t)
|
||
|
|
|
||
|
|
.. math:: y' = f_2(x) g_2(y) \phi(x,y,t)
|
||
|
|
|
||
|
|
First integral:
|
||
|
|
|
||
|
|
.. math:: \int \frac{f_2(x)}{f_1(x)} \,dx - \int \frac{g_1(y)}{g_2(y)} \,dy = C
|
||
|
|
|
||
|
|
where `C` is an arbitrary constant.
|
||
|
|
|
||
|
|
On solving the first integral for `x` (resp., `y` ) and on substituting the
|
||
|
|
resulting expression into either equation of the original solution, one
|
||
|
|
arrives at a first-order equation for determining `y` (resp., `x` ).
|
||
|
|
|
||
|
|
"""
|
||
|
|
C1, C2 = get_numbered_constants(eq, num=2)
|
||
|
|
u, v = symbols('u, v')
|
||
|
|
U, V = symbols('U, V', cls=Function)
|
||
|
|
f = Wild('f')
|
||
|
|
g = Wild('g')
|
||
|
|
f1 = Wild('f1', exclude=[v,t])
|
||
|
|
f2 = Wild('f2', exclude=[v,t])
|
||
|
|
g1 = Wild('g1', exclude=[u,t])
|
||
|
|
g2 = Wild('g2', exclude=[u,t])
|
||
|
|
r1 = eq[0].match(diff(x(t),t) - f)
|
||
|
|
r2 = eq[1].match(diff(y(t),t) - g)
|
||
|
|
num, den = (
|
||
|
|
(r1[f].subs(x(t),u).subs(y(t),v))/
|
||
|
|
(r2[g].subs(x(t),u).subs(y(t),v))).as_numer_denom()
|
||
|
|
R1 = num.match(f1*g1)
|
||
|
|
R2 = den.match(f2*g2)
|
||
|
|
phi = (r1[f].subs(x(t),u).subs(y(t),v))/num
|
||
|
|
F1 = R1[f1]; F2 = R2[f2]
|
||
|
|
G1 = R1[g1]; G2 = R2[g2]
|
||
|
|
sol1r = solve(Integral(F2/F1, u).doit() - Integral(G1/G2,v).doit() - C1, u)
|
||
|
|
sol2r = solve(Integral(F2/F1, u).doit() - Integral(G1/G2,v).doit() - C1, v)
|
||
|
|
sol = []
|
||
|
|
for sols in sol1r:
|
||
|
|
sol.append(Eq(y(t), dsolve(diff(V(t),t) - F2.subs(u,sols).subs(v,V(t))*G2.subs(v,V(t))*phi.subs(u,sols).subs(v,V(t))).rhs))
|
||
|
|
for sols in sol2r:
|
||
|
|
sol.append(Eq(x(t), dsolve(diff(U(t),t) - F1.subs(u,U(t))*G1.subs(v,sols).subs(u,U(t))*phi.subs(v,sols).subs(u,U(t))).rhs))
|
||
|
|
return set(sol)
|
||
|
|
|
||
|
|
def _nonlinear_2eq_order1_type5(func, t, eq):
|
||
|
|
r"""
|
||
|
|
Clairaut system of ODEs
|
||
|
|
|
||
|
|
.. math:: x = t x' + F(x',y')
|
||
|
|
|
||
|
|
.. math:: y = t y' + G(x',y')
|
||
|
|
|
||
|
|
The following are solutions of the system
|
||
|
|
|
||
|
|
`(i)` straight lines:
|
||
|
|
|
||
|
|
.. math:: x = C_1 t + F(C_1, C_2), y = C_2 t + G(C_1, C_2)
|
||
|
|
|
||
|
|
where `C_1` and `C_2` are arbitrary constants;
|
||
|
|
|
||
|
|
`(ii)` envelopes of the above lines;
|
||
|
|
|
||
|
|
`(iii)` continuously differentiable lines made up from segments of the lines
|
||
|
|
`(i)` and `(ii)`.
|
||
|
|
|
||
|
|
"""
|
||
|
|
C1, C2 = get_numbered_constants(eq, num=2)
|
||
|
|
f = Wild('f')
|
||
|
|
g = Wild('g')
|
||
|
|
def check_type(x, y):
|
||
|
|
r1 = eq[0].match(t*diff(x(t),t) - x(t) + f)
|
||
|
|
r2 = eq[1].match(t*diff(y(t),t) - y(t) + g)
|
||
|
|
if not (r1 and r2):
|
||
|
|
r1 = eq[0].match(diff(x(t),t) - x(t)/t + f/t)
|
||
|
|
r2 = eq[1].match(diff(y(t),t) - y(t)/t + g/t)
|
||
|
|
if not (r1 and r2):
|
||
|
|
r1 = (-eq[0]).match(t*diff(x(t),t) - x(t) + f)
|
||
|
|
r2 = (-eq[1]).match(t*diff(y(t),t) - y(t) + g)
|
||
|
|
if not (r1 and r2):
|
||
|
|
r1 = (-eq[0]).match(diff(x(t),t) - x(t)/t + f/t)
|
||
|
|
r2 = (-eq[1]).match(diff(y(t),t) - y(t)/t + g/t)
|
||
|
|
return [r1, r2]
|
||
|
|
for func_ in func:
|
||
|
|
if isinstance(func_, list):
|
||
|
|
x = func[0][0].func
|
||
|
|
y = func[0][1].func
|
||
|
|
[r1, r2] = check_type(x, y)
|
||
|
|
if not (r1 and r2):
|
||
|
|
[r1, r2] = check_type(y, x)
|
||
|
|
x, y = y, x
|
||
|
|
x1 = diff(x(t),t); y1 = diff(y(t),t)
|
||
|
|
return {Eq(x(t), C1*t + r1[f].subs(x1,C1).subs(y1,C2)), Eq(y(t), C2*t + r2[g].subs(x1,C1).subs(y1,C2))}
|
||
|
|
|
||
|
|
def sysode_nonlinear_3eq_order1(match_):
|
||
|
|
x = match_['func'][0].func
|
||
|
|
y = match_['func'][1].func
|
||
|
|
z = match_['func'][2].func
|
||
|
|
eq = match_['eq']
|
||
|
|
t = list(list(eq[0].atoms(Derivative))[0].atoms(Symbol))[0]
|
||
|
|
if match_['type_of_equation'] == 'type1':
|
||
|
|
sol = _nonlinear_3eq_order1_type1(x, y, z, t, eq)
|
||
|
|
if match_['type_of_equation'] == 'type2':
|
||
|
|
sol = _nonlinear_3eq_order1_type2(x, y, z, t, eq)
|
||
|
|
if match_['type_of_equation'] == 'type3':
|
||
|
|
sol = _nonlinear_3eq_order1_type3(x, y, z, t, eq)
|
||
|
|
if match_['type_of_equation'] == 'type4':
|
||
|
|
sol = _nonlinear_3eq_order1_type4(x, y, z, t, eq)
|
||
|
|
if match_['type_of_equation'] == 'type5':
|
||
|
|
sol = _nonlinear_3eq_order1_type5(x, y, z, t, eq)
|
||
|
|
return sol
|
||
|
|
|
||
|
|
def _nonlinear_3eq_order1_type1(x, y, z, t, eq):
|
||
|
|
r"""
|
||
|
|
Equations:
|
||
|
|
|
||
|
|
.. math:: a x' = (b - c) y z, \enspace b y' = (c - a) z x, \enspace c z' = (a - b) x y
|
||
|
|
|
||
|
|
First Integrals:
|
||
|
|
|
||
|
|
.. math:: a x^{2} + b y^{2} + c z^{2} = C_1
|
||
|
|
|
||
|
|
.. math:: a^{2} x^{2} + b^{2} y^{2} + c^{2} z^{2} = C_2
|
||
|
|
|
||
|
|
where `C_1` and `C_2` are arbitrary constants. On solving the integrals for `y` and
|
||
|
|
`z` and on substituting the resulting expressions into the first equation of the
|
||
|
|
system, we arrives at a separable first-order equation on `x`. Similarly doing that
|
||
|
|
for other two equations, we will arrive at first order equation on `y` and `z` too.
|
||
|
|
|
||
|
|
References
|
||
|
|
==========
|
||
|
|
-https://eqworld.ipmnet.ru/en/solutions/sysode/sode0401.pdf
|
||
|
|
|
||
|
|
"""
|
||
|
|
C1, C2 = get_numbered_constants(eq, num=2)
|
||
|
|
u, v, w = symbols('u, v, w')
|
||
|
|
p = Wild('p', exclude=[x(t), y(t), z(t), t])
|
||
|
|
q = Wild('q', exclude=[x(t), y(t), z(t), t])
|
||
|
|
s = Wild('s', exclude=[x(t), y(t), z(t), t])
|
||
|
|
r = (diff(x(t),t) - eq[0]).match(p*y(t)*z(t))
|
||
|
|
r.update((diff(y(t),t) - eq[1]).match(q*z(t)*x(t)))
|
||
|
|
r.update((diff(z(t),t) - eq[2]).match(s*x(t)*y(t)))
|
||
|
|
n1, d1 = r[p].as_numer_denom()
|
||
|
|
n2, d2 = r[q].as_numer_denom()
|
||
|
|
n3, d3 = r[s].as_numer_denom()
|
||
|
|
val = solve([n1*u-d1*v+d1*w, d2*u+n2*v-d2*w, d3*u-d3*v-n3*w],[u,v])
|
||
|
|
vals = [val[v], val[u]]
|
||
|
|
c = lcm(vals[0].as_numer_denom()[1], vals[1].as_numer_denom()[1])
|
||
|
|
b = vals[0].subs(w, c)
|
||
|
|
a = vals[1].subs(w, c)
|
||
|
|
y_x = sqrt(((c*C1-C2) - a*(c-a)*x(t)**2)/(b*(c-b)))
|
||
|
|
z_x = sqrt(((b*C1-C2) - a*(b-a)*x(t)**2)/(c*(b-c)))
|
||
|
|
z_y = sqrt(((a*C1-C2) - b*(a-b)*y(t)**2)/(c*(a-c)))
|
||
|
|
x_y = sqrt(((c*C1-C2) - b*(c-b)*y(t)**2)/(a*(c-a)))
|
||
|
|
x_z = sqrt(((b*C1-C2) - c*(b-c)*z(t)**2)/(a*(b-a)))
|
||
|
|
y_z = sqrt(((a*C1-C2) - c*(a-c)*z(t)**2)/(b*(a-b)))
|
||
|
|
sol1 = dsolve(a*diff(x(t),t) - (b-c)*y_x*z_x)
|
||
|
|
sol2 = dsolve(b*diff(y(t),t) - (c-a)*z_y*x_y)
|
||
|
|
sol3 = dsolve(c*diff(z(t),t) - (a-b)*x_z*y_z)
|
||
|
|
return [sol1, sol2, sol3]
|
||
|
|
|
||
|
|
|
||
|
|
def _nonlinear_3eq_order1_type2(x, y, z, t, eq):
|
||
|
|
r"""
|
||
|
|
Equations:
|
||
|
|
|
||
|
|
.. math:: a x' = (b - c) y z f(x, y, z, t)
|
||
|
|
|
||
|
|
.. math:: b y' = (c - a) z x f(x, y, z, t)
|
||
|
|
|
||
|
|
.. math:: c z' = (a - b) x y f(x, y, z, t)
|
||
|
|
|
||
|
|
First Integrals:
|
||
|
|
|
||
|
|
.. math:: a x^{2} + b y^{2} + c z^{2} = C_1
|
||
|
|
|
||
|
|
.. math:: a^{2} x^{2} + b^{2} y^{2} + c^{2} z^{2} = C_2
|
||
|
|
|
||
|
|
where `C_1` and `C_2` are arbitrary constants. On solving the integrals for `y` and
|
||
|
|
`z` and on substituting the resulting expressions into the first equation of the
|
||
|
|
system, we arrives at a first-order differential equations on `x`. Similarly doing
|
||
|
|
that for other two equations we will arrive at first order equation on `y` and `z`.
|
||
|
|
|
||
|
|
References
|
||
|
|
==========
|
||
|
|
-https://eqworld.ipmnet.ru/en/solutions/sysode/sode0402.pdf
|
||
|
|
|
||
|
|
"""
|
||
|
|
C1, C2 = get_numbered_constants(eq, num=2)
|
||
|
|
u, v, w = symbols('u, v, w')
|
||
|
|
p = Wild('p', exclude=[x(t), y(t), z(t), t])
|
||
|
|
q = Wild('q', exclude=[x(t), y(t), z(t), t])
|
||
|
|
s = Wild('s', exclude=[x(t), y(t), z(t), t])
|
||
|
|
f = Wild('f')
|
||
|
|
r1 = (diff(x(t),t) - eq[0]).match(y(t)*z(t)*f)
|
||
|
|
r = collect_const(r1[f]).match(p*f)
|
||
|
|
r.update(((diff(y(t),t) - eq[1])/r[f]).match(q*z(t)*x(t)))
|
||
|
|
r.update(((diff(z(t),t) - eq[2])/r[f]).match(s*x(t)*y(t)))
|
||
|
|
n1, d1 = r[p].as_numer_denom()
|
||
|
|
n2, d2 = r[q].as_numer_denom()
|
||
|
|
n3, d3 = r[s].as_numer_denom()
|
||
|
|
val = solve([n1*u-d1*v+d1*w, d2*u+n2*v-d2*w, -d3*u+d3*v+n3*w],[u,v])
|
||
|
|
vals = [val[v], val[u]]
|
||
|
|
c = lcm(vals[0].as_numer_denom()[1], vals[1].as_numer_denom()[1])
|
||
|
|
a = vals[0].subs(w, c)
|
||
|
|
b = vals[1].subs(w, c)
|
||
|
|
y_x = sqrt(((c*C1-C2) - a*(c-a)*x(t)**2)/(b*(c-b)))
|
||
|
|
z_x = sqrt(((b*C1-C2) - a*(b-a)*x(t)**2)/(c*(b-c)))
|
||
|
|
z_y = sqrt(((a*C1-C2) - b*(a-b)*y(t)**2)/(c*(a-c)))
|
||
|
|
x_y = sqrt(((c*C1-C2) - b*(c-b)*y(t)**2)/(a*(c-a)))
|
||
|
|
x_z = sqrt(((b*C1-C2) - c*(b-c)*z(t)**2)/(a*(b-a)))
|
||
|
|
y_z = sqrt(((a*C1-C2) - c*(a-c)*z(t)**2)/(b*(a-b)))
|
||
|
|
sol1 = dsolve(a*diff(x(t),t) - (b-c)*y_x*z_x*r[f])
|
||
|
|
sol2 = dsolve(b*diff(y(t),t) - (c-a)*z_y*x_y*r[f])
|
||
|
|
sol3 = dsolve(c*diff(z(t),t) - (a-b)*x_z*y_z*r[f])
|
||
|
|
return [sol1, sol2, sol3]
|
||
|
|
|
||
|
|
def _nonlinear_3eq_order1_type3(x, y, z, t, eq):
|
||
|
|
r"""
|
||
|
|
Equations:
|
||
|
|
|
||
|
|
.. math:: x' = c F_2 - b F_3, \enspace y' = a F_3 - c F_1, \enspace z' = b F_1 - a F_2
|
||
|
|
|
||
|
|
where `F_n = F_n(x, y, z, t)`.
|
||
|
|
|
||
|
|
1. First Integral:
|
||
|
|
|
||
|
|
.. math:: a x + b y + c z = C_1,
|
||
|
|
|
||
|
|
where C is an arbitrary constant.
|
||
|
|
|
||
|
|
2. If we assume function `F_n` to be independent of `t`,i.e, `F_n` = `F_n (x, y, z)`
|
||
|
|
Then, on eliminating `t` and `z` from the first two equation of the system, one
|
||
|
|
arrives at the first-order equation
|
||
|
|
|
||
|
|
.. math:: \frac{dy}{dx} = \frac{a F_3 (x, y, z) - c F_1 (x, y, z)}{c F_2 (x, y, z) -
|
||
|
|
b F_3 (x, y, z)}
|
||
|
|
|
||
|
|
where `z = \frac{1}{c} (C_1 - a x - b y)`
|
||
|
|
|
||
|
|
References
|
||
|
|
==========
|
||
|
|
-https://eqworld.ipmnet.ru/en/solutions/sysode/sode0404.pdf
|
||
|
|
|
||
|
|
"""
|
||
|
|
C1 = get_numbered_constants(eq, num=1)
|
||
|
|
u, v, w = symbols('u, v, w')
|
||
|
|
fu, fv, fw = symbols('u, v, w', cls=Function)
|
||
|
|
p = Wild('p', exclude=[x(t), y(t), z(t), t])
|
||
|
|
q = Wild('q', exclude=[x(t), y(t), z(t), t])
|
||
|
|
s = Wild('s', exclude=[x(t), y(t), z(t), t])
|
||
|
|
F1, F2, F3 = symbols('F1, F2, F3', cls=Wild)
|
||
|
|
r1 = (diff(x(t), t) - eq[0]).match(F2-F3)
|
||
|
|
r = collect_const(r1[F2]).match(s*F2)
|
||
|
|
r.update(collect_const(r1[F3]).match(q*F3))
|
||
|
|
if eq[1].has(r[F2]) and not eq[1].has(r[F3]):
|
||
|
|
r[F2], r[F3] = r[F3], r[F2]
|
||
|
|
r[s], r[q] = -r[q], -r[s]
|
||
|
|
r.update((diff(y(t), t) - eq[1]).match(p*r[F3] - r[s]*F1))
|
||
|
|
a = r[p]; b = r[q]; c = r[s]
|
||
|
|
F1 = r[F1].subs(x(t), u).subs(y(t),v).subs(z(t), w)
|
||
|
|
F2 = r[F2].subs(x(t), u).subs(y(t),v).subs(z(t), w)
|
||
|
|
F3 = r[F3].subs(x(t), u).subs(y(t),v).subs(z(t), w)
|
||
|
|
z_xy = (C1-a*u-b*v)/c
|
||
|
|
y_zx = (C1-a*u-c*w)/b
|
||
|
|
x_yz = (C1-b*v-c*w)/a
|
||
|
|
y_x = dsolve(diff(fv(u),u) - ((a*F3-c*F1)/(c*F2-b*F3)).subs(w,z_xy).subs(v,fv(u))).rhs
|
||
|
|
z_x = dsolve(diff(fw(u),u) - ((b*F1-a*F2)/(c*F2-b*F3)).subs(v,y_zx).subs(w,fw(u))).rhs
|
||
|
|
z_y = dsolve(diff(fw(v),v) - ((b*F1-a*F2)/(a*F3-c*F1)).subs(u,x_yz).subs(w,fw(v))).rhs
|
||
|
|
x_y = dsolve(diff(fu(v),v) - ((c*F2-b*F3)/(a*F3-c*F1)).subs(w,z_xy).subs(u,fu(v))).rhs
|
||
|
|
y_z = dsolve(diff(fv(w),w) - ((a*F3-c*F1)/(b*F1-a*F2)).subs(u,x_yz).subs(v,fv(w))).rhs
|
||
|
|
x_z = dsolve(diff(fu(w),w) - ((c*F2-b*F3)/(b*F1-a*F2)).subs(v,y_zx).subs(u,fu(w))).rhs
|
||
|
|
sol1 = dsolve(diff(fu(t),t) - (c*F2 - b*F3).subs(v,y_x).subs(w,z_x).subs(u,fu(t))).rhs
|
||
|
|
sol2 = dsolve(diff(fv(t),t) - (a*F3 - c*F1).subs(u,x_y).subs(w,z_y).subs(v,fv(t))).rhs
|
||
|
|
sol3 = dsolve(diff(fw(t),t) - (b*F1 - a*F2).subs(u,x_z).subs(v,y_z).subs(w,fw(t))).rhs
|
||
|
|
return [sol1, sol2, sol3]
|
||
|
|
|
||
|
|
def _nonlinear_3eq_order1_type4(x, y, z, t, eq):
|
||
|
|
r"""
|
||
|
|
Equations:
|
||
|
|
|
||
|
|
.. math:: x' = c z F_2 - b y F_3, \enspace y' = a x F_3 - c z F_1, \enspace z' = b y F_1 - a x F_2
|
||
|
|
|
||
|
|
where `F_n = F_n (x, y, z, t)`
|
||
|
|
|
||
|
|
1. First integral:
|
||
|
|
|
||
|
|
.. math:: a x^{2} + b y^{2} + c z^{2} = C_1
|
||
|
|
|
||
|
|
where `C` is an arbitrary constant.
|
||
|
|
|
||
|
|
2. Assuming the function `F_n` is independent of `t`: `F_n = F_n (x, y, z)`. Then on
|
||
|
|
eliminating `t` and `z` from the first two equations of the system, one arrives at
|
||
|
|
the first-order equation
|
||
|
|
|
||
|
|
.. math:: \frac{dy}{dx} = \frac{a x F_3 (x, y, z) - c z F_1 (x, y, z)}
|
||
|
|
{c z F_2 (x, y, z) - b y F_3 (x, y, z)}
|
||
|
|
|
||
|
|
where `z = \pm \sqrt{\frac{1}{c} (C_1 - a x^{2} - b y^{2})}`
|
||
|
|
|
||
|
|
References
|
||
|
|
==========
|
||
|
|
-https://eqworld.ipmnet.ru/en/solutions/sysode/sode0405.pdf
|
||
|
|
|
||
|
|
"""
|
||
|
|
C1 = get_numbered_constants(eq, num=1)
|
||
|
|
u, v, w = symbols('u, v, w')
|
||
|
|
p = Wild('p', exclude=[x(t), y(t), z(t), t])
|
||
|
|
q = Wild('q', exclude=[x(t), y(t), z(t), t])
|
||
|
|
s = Wild('s', exclude=[x(t), y(t), z(t), t])
|
||
|
|
F1, F2, F3 = symbols('F1, F2, F3', cls=Wild)
|
||
|
|
r1 = eq[0].match(diff(x(t),t) - z(t)*F2 + y(t)*F3)
|
||
|
|
r = collect_const(r1[F2]).match(s*F2)
|
||
|
|
r.update(collect_const(r1[F3]).match(q*F3))
|
||
|
|
if eq[1].has(r[F2]) and not eq[1].has(r[F3]):
|
||
|
|
r[F2], r[F3] = r[F3], r[F2]
|
||
|
|
r[s], r[q] = -r[q], -r[s]
|
||
|
|
r.update((diff(y(t),t) - eq[1]).match(p*x(t)*r[F3] - r[s]*z(t)*F1))
|
||
|
|
a = r[p]; b = r[q]; c = r[s]
|
||
|
|
F1 = r[F1].subs(x(t),u).subs(y(t),v).subs(z(t),w)
|
||
|
|
F2 = r[F2].subs(x(t),u).subs(y(t),v).subs(z(t),w)
|
||
|
|
F3 = r[F3].subs(x(t),u).subs(y(t),v).subs(z(t),w)
|
||
|
|
x_yz = sqrt((C1 - b*v**2 - c*w**2)/a)
|
||
|
|
y_zx = sqrt((C1 - c*w**2 - a*u**2)/b)
|
||
|
|
z_xy = sqrt((C1 - a*u**2 - b*v**2)/c)
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y_x = dsolve(diff(v(u),u) - ((a*u*F3-c*w*F1)/(c*w*F2-b*v*F3)).subs(w,z_xy).subs(v,v(u))).rhs
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z_x = dsolve(diff(w(u),u) - ((b*v*F1-a*u*F2)/(c*w*F2-b*v*F3)).subs(v,y_zx).subs(w,w(u))).rhs
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z_y = dsolve(diff(w(v),v) - ((b*v*F1-a*u*F2)/(a*u*F3-c*w*F1)).subs(u,x_yz).subs(w,w(v))).rhs
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x_y = dsolve(diff(u(v),v) - ((c*w*F2-b*v*F3)/(a*u*F3-c*w*F1)).subs(w,z_xy).subs(u,u(v))).rhs
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y_z = dsolve(diff(v(w),w) - ((a*u*F3-c*w*F1)/(b*v*F1-a*u*F2)).subs(u,x_yz).subs(v,v(w))).rhs
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x_z = dsolve(diff(u(w),w) - ((c*w*F2-b*v*F3)/(b*v*F1-a*u*F2)).subs(v,y_zx).subs(u,u(w))).rhs
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sol1 = dsolve(diff(u(t),t) - (c*w*F2 - b*v*F3).subs(v,y_x).subs(w,z_x).subs(u,u(t))).rhs
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sol2 = dsolve(diff(v(t),t) - (a*u*F3 - c*w*F1).subs(u,x_y).subs(w,z_y).subs(v,v(t))).rhs
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sol3 = dsolve(diff(w(t),t) - (b*v*F1 - a*u*F2).subs(u,x_z).subs(v,y_z).subs(w,w(t))).rhs
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return [sol1, sol2, sol3]
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def _nonlinear_3eq_order1_type5(x, y, z, t, eq):
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r"""
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.. math:: x' = x (c F_2 - b F_3), \enspace y' = y (a F_3 - c F_1), \enspace z' = z (b F_1 - a F_2)
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where `F_n = F_n (x, y, z, t)` and are arbitrary functions.
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First Integral:
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.. math:: \left|x\right|^{a} \left|y\right|^{b} \left|z\right|^{c} = C_1
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where `C` is an arbitrary constant. If the function `F_n` is independent of `t`,
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then, by eliminating `t` and `z` from the first two equations of the system, one
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arrives at a first-order equation.
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References
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|
|
==========
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|
-https://eqworld.ipmnet.ru/en/solutions/sysode/sode0406.pdf
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"""
|
||
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|
C1 = get_numbered_constants(eq, num=1)
|
||
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u, v, w = symbols('u, v, w')
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||
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fu, fv, fw = symbols('u, v, w', cls=Function)
|
||
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p = Wild('p', exclude=[x(t), y(t), z(t), t])
|
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q = Wild('q', exclude=[x(t), y(t), z(t), t])
|
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|
|
s = Wild('s', exclude=[x(t), y(t), z(t), t])
|
||
|
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F1, F2, F3 = symbols('F1, F2, F3', cls=Wild)
|
||
|
|
r1 = eq[0].match(diff(x(t), t) - x(t)*F2 + x(t)*F3)
|
||
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|
r = collect_const(r1[F2]).match(s*F2)
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||
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r.update(collect_const(r1[F3]).match(q*F3))
|
||
|
|
if eq[1].has(r[F2]) and not eq[1].has(r[F3]):
|
||
|
|
r[F2], r[F3] = r[F3], r[F2]
|
||
|
|
r[s], r[q] = -r[q], -r[s]
|
||
|
|
r.update((diff(y(t), t) - eq[1]).match(y(t)*(p*r[F3] - r[s]*F1)))
|
||
|
|
a = r[p]; b = r[q]; c = r[s]
|
||
|
|
F1 = r[F1].subs(x(t), u).subs(y(t), v).subs(z(t), w)
|
||
|
|
F2 = r[F2].subs(x(t), u).subs(y(t), v).subs(z(t), w)
|
||
|
|
F3 = r[F3].subs(x(t), u).subs(y(t), v).subs(z(t), w)
|
||
|
|
x_yz = (C1*v**-b*w**-c)**-a
|
||
|
|
y_zx = (C1*w**-c*u**-a)**-b
|
||
|
|
z_xy = (C1*u**-a*v**-b)**-c
|
||
|
|
y_x = dsolve(diff(fv(u), u) - ((v*(a*F3 - c*F1))/(u*(c*F2 - b*F3))).subs(w, z_xy).subs(v, fv(u))).rhs
|
||
|
|
z_x = dsolve(diff(fw(u), u) - ((w*(b*F1 - a*F2))/(u*(c*F2 - b*F3))).subs(v, y_zx).subs(w, fw(u))).rhs
|
||
|
|
z_y = dsolve(diff(fw(v), v) - ((w*(b*F1 - a*F2))/(v*(a*F3 - c*F1))).subs(u, x_yz).subs(w, fw(v))).rhs
|
||
|
|
x_y = dsolve(diff(fu(v), v) - ((u*(c*F2 - b*F3))/(v*(a*F3 - c*F1))).subs(w, z_xy).subs(u, fu(v))).rhs
|
||
|
|
y_z = dsolve(diff(fv(w), w) - ((v*(a*F3 - c*F1))/(w*(b*F1 - a*F2))).subs(u, x_yz).subs(v, fv(w))).rhs
|
||
|
|
x_z = dsolve(diff(fu(w), w) - ((u*(c*F2 - b*F3))/(w*(b*F1 - a*F2))).subs(v, y_zx).subs(u, fu(w))).rhs
|
||
|
|
sol1 = dsolve(diff(fu(t), t) - (u*(c*F2 - b*F3)).subs(v, y_x).subs(w, z_x).subs(u, fu(t))).rhs
|
||
|
|
sol2 = dsolve(diff(fv(t), t) - (v*(a*F3 - c*F1)).subs(u, x_y).subs(w, z_y).subs(v, fv(t))).rhs
|
||
|
|
sol3 = dsolve(diff(fw(t), t) - (w*(b*F1 - a*F2)).subs(u, x_z).subs(v, y_z).subs(w, fw(t))).rhs
|
||
|
|
return [sol1, sol2, sol3]
|
||
|
|
|
||
|
|
|
||
|
|
#This import is written at the bottom to avoid circular imports.
|
||
|
|
from .single import SingleODEProblem, SingleODESolver, solver_map
|