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ODE
===
.. note::
For a beginner-friendly guide focused on solving ODEs, refer to
:ref:`solving-guide-ode`.
.. module::sympy.solvers.ode
.. automodule:: sympy.solvers.ode
User Functions
--------------
These are functions that are imported into the global namespace with ``from
sympy import *``. These functions (unlike `Hint Functions`_, below) are
intended for use by ordinary users of SymPy.
.. autofunction:: sympy.solvers.ode::dsolve
.. autofunction:: sympy.solvers.ode.systems::dsolve_system
.. autofunction:: sympy.solvers.ode::classify_ode
.. autofunction:: sympy.solvers.ode::checkodesol
.. autofunction:: sympy.solvers.ode::homogeneous_order
.. autofunction:: sympy.solvers.ode::infinitesimals
.. autofunction:: sympy.solvers.ode::checkinfsol
.. autofunction:: sympy.solvers.ode::constantsimp
.. _hints:
Hint Functions
--------------
These functions are intended for internal use by
:py:meth:`~sympy.solvers.ode.dsolve` and others. Unlike `User Functions`_,
above, these are not intended for every-day use by ordinary SymPy users.
Instead, functions such as :py:meth:`~sympy.solvers.ode.dsolve` should be used.
Nonetheless, these functions contain useful information in their docstrings on
the various ODE solving methods. For this reason, they are documented here.
.. autodata:: sympy.solvers.ode::allhints
.. autofunction:: sympy.solvers.ode.ode::odesimp
.. autofunction:: sympy.solvers.ode.ode::constant_renumber
.. autofunction:: sympy.solvers.ode.ode::ode_sol_simplicity
.. autoclass:: sympy.solvers.ode.single::Factorable
:members:
.. autoclass:: sympy.solvers.ode.single.FirstExact
:members:
.. autoclass:: sympy.solvers.ode.single::HomogeneousCoeffBest
:members:
.. autoclass:: sympy.solvers.ode.single::HomogeneousCoeffSubsDepDivIndep
:members:
.. autoclass:: sympy.solvers.ode.single::HomogeneousCoeffSubsIndepDivDep
:members:
.. autoclass:: sympy.solvers.ode.single::FirstLinear
:members:
.. autoclass:: sympy.solvers.ode.single::RationalRiccati
:members:
.. autoclass:: sympy.solvers.ode.single::SecondLinearAiry
:members:
.. autoclass:: sympy.solvers.ode.single::SecondLinearBessel
:members:
.. autoclass:: sympy.solvers.ode.single::Bernoulli
:members:
.. autoclass:: sympy.solvers.ode.single::Liouville
:members:
.. autoclass:: sympy.solvers.ode.single::RiccatiSpecial
:members:
.. autoclass:: sympy.solvers.ode.single::NthLinearConstantCoeffHomogeneous
:members:
.. autoclass:: sympy.solvers.ode.single::NthLinearConstantCoeffUndeterminedCoefficients
:members:
.. autoclass:: sympy.solvers.ode.single::NthLinearConstantCoeffVariationOfParameters
:members:
.. autoclass:: sympy.solvers.ode.single::NthLinearEulerEqHomogeneous
:members:
.. autoclass:: sympy.solvers.ode.single::NthLinearEulerEqNonhomogeneousVariationOfParameters
:members:
.. autoclass:: sympy.solvers.ode.single::NthLinearEulerEqNonhomogeneousUndeterminedCoefficients
:members:
.. autoclass:: sympy.solvers.ode.single::NthAlgebraic
:members:
.. autoclass:: sympy.solvers.ode.single::NthOrderReducible
:members:
.. autoclass:: sympy.solvers.ode.single::Separable
:members:
.. autoclass:: sympy.solvers.ode.single::AlmostLinear
:members:
.. autoclass:: sympy.solvers.ode.single::LinearCoefficients
:members:
.. autoclass:: sympy.solvers.ode.single::SeparableReduced
:members:
.. autoclass:: sympy.solvers.ode.single::LieGroup
:members:
.. autoclass:: sympy.solvers.ode.single::SecondHypergeometric
:members:
.. autofunction:: sympy.solvers.ode.ode::ode_1st_power_series
.. autofunction:: sympy.solvers.ode.ode::ode_2nd_power_series_ordinary
.. autofunction:: sympy.solvers.ode.ode::ode_2nd_power_series_regular
Lie heuristics
--------------
These functions are intended for internal use of the Lie Group Solver.
Nonetheless, they contain useful information in their docstrings on the
algorithms implemented for the various heuristics.
.. autofunction:: sympy.solvers.ode.lie_group::lie_heuristic_abaco1_simple
.. autofunction:: sympy.solvers.ode.lie_group::lie_heuristic_abaco1_product
.. autofunction:: sympy.solvers.ode.lie_group::lie_heuristic_bivariate
.. autofunction:: sympy.solvers.ode.lie_group::lie_heuristic_chi
.. autofunction:: sympy.solvers.ode.lie_group::lie_heuristic_abaco2_similar
.. autofunction:: sympy.solvers.ode.lie_group::lie_heuristic_function_sum
.. autofunction:: sympy.solvers.ode.lie_group::lie_heuristic_abaco2_unique_unknown
.. autofunction:: sympy.solvers.ode.lie_group::lie_heuristic_abaco2_unique_general
.. autofunction:: sympy.solvers.ode.lie_group::lie_heuristic_linear
Rational Riccati Solver
-----------------------
These functions are intended for internal use to solve a first order Riccati
differential equation with at least one rational particular solution.
.. autofunction:: sympy.solvers.ode.riccati::riccati_normal
.. autofunction:: sympy.solvers.ode.riccati::riccati_inverse_normal
.. autofunction:: sympy.solvers.ode.riccati::riccati_reduced
.. autofunction:: sympy.solvers.ode.riccati::construct_c
.. autofunction:: sympy.solvers.ode.riccati::construct_d
.. autofunction:: sympy.solvers.ode.riccati::rational_laurent_series
.. autofunction:: sympy.solvers.ode.riccati::compute_m_ybar
.. autofunction:: sympy.solvers.ode.riccati::solve_aux_eq
.. autofunction:: sympy.solvers.ode.riccati::remove_redundant_sols
.. autofunction:: sympy.solvers.ode.riccati::get_gen_sol_from_part_sol
.. autofunction:: sympy.solvers.ode.riccati::solve_riccati
System of ODEs
--------------
These functions are intended for internal use by
:py:meth:`~sympy.solvers.ode.dsolve` for system of differential equations.
.. autofunction:: sympy.solvers.ode.ode::_linear_2eq_order1_type6
.. autofunction:: sympy.solvers.ode.ode::_linear_2eq_order1_type7
.. autofunction:: sympy.solvers.ode.systems::linear_ode_to_matrix
.. autofunction:: sympy.solvers.ode.systems::canonical_odes
.. autofunction:: sympy.solvers.ode.systems::linodesolve_type
.. autofunction:: sympy.solvers.ode.systems::matrix_exp_jordan_form
.. autofunction:: sympy.solvers.ode.systems::matrix_exp
.. autofunction:: sympy.solvers.ode.systems::linodesolve
.. autofunction:: sympy.solvers.ode.ode::_nonlinear_2eq_order1_type1
.. autofunction:: sympy.solvers.ode.ode::_nonlinear_2eq_order1_type2
.. autofunction:: sympy.solvers.ode.ode::_nonlinear_2eq_order1_type3
.. autofunction:: sympy.solvers.ode.ode::_nonlinear_2eq_order1_type4
.. autofunction:: sympy.solvers.ode.ode::_nonlinear_2eq_order1_type5
.. autofunction:: sympy.solvers.ode.ode::_nonlinear_3eq_order1_type1
.. autofunction:: sympy.solvers.ode.ode::_nonlinear_3eq_order1_type2
.. autofunction:: sympy.solvers.ode.ode::_nonlinear_3eq_order1_type3
.. autofunction:: sympy.solvers.ode.ode::_nonlinear_3eq_order1_type4
.. autofunction:: sympy.solvers.ode.ode::_nonlinear_3eq_order1_type5
Information on the ode module
-----------------------------
.. automodule:: sympy.solvers.ode.ode
These functions are not intended for end-user use.
.. autofunction:: sympy.solvers.ode.ode::_handle_Integral
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