.. _ode-docs:

ODE
===

.. module::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.

:func:`dsolve`
^^^^^^^^^^^^^^
.. autofunction:: sympy.solvers.ode.dsolve

:func:`classify_ode`
^^^^^^^^^^^^^^^^^^^^
.. autofunction:: sympy.solvers.ode.classify_ode

:func:`checkodesol`
^^^^^^^^^^^^^^^^^^^
.. autofunction:: sympy.solvers.ode.checkodesol

:func:`homogeneous_order`
^^^^^^^^^^^^^^^^^^^^^^^^^
.. autofunction:: sympy.solvers.ode.homogeneous_order

:func:`infinitesimals`
^^^^^^^^^^^^^^^^^^^^^^
.. autofunction:: sympy.solvers.ode.infinitesimals

:func:`checkinfsol`
^^^^^^^^^^^^^^^^^^^
.. autofunction:: sympy.solvers.ode.checkinfsol

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.

:obj:`allhints`
^^^^^^^^^^^^^^^
.. autodata:: sympy.solvers.ode.allhints

:obj:`odesimp`
^^^^^^^^^^^^^^
.. autofunction:: sympy.solvers.ode.odesimp

:obj:`constant_renumber`
^^^^^^^^^^^^^^^^^^^^^^^^
.. autofunction:: sympy.solvers.ode.constant_renumber

:obj:`constantsimp`
^^^^^^^^^^^^^^^^^^^
.. autofunction:: sympy.solvers.ode.constantsimp

:obj:`sol_simplicity`
^^^^^^^^^^^^^^^^^^^^^^^^^
.. autofunction:: sympy.solvers.ode.ode_sol_simplicity

:obj:`1st_exact`
^^^^^^^^^^^^^^^^
.. autofunction:: sympy.solvers.ode.ode_1st_exact

:obj:`1st_homogeneous_coeff_best`
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
.. autofunction:: sympy.solvers.ode.ode_1st_homogeneous_coeff_best

:obj:`1st_homogeneous_coeff_subs_dep_div_indep`
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
.. autofunction:: sympy.solvers.ode.ode_1st_homogeneous_coeff_subs_dep_div_indep

:obj:`1st_homogeneous_coeff_subs_indep_div_dep`
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
.. autofunction:: sympy.solvers.ode.ode_1st_homogeneous_coeff_subs_indep_div_dep

:obj:`1st_linear`
^^^^^^^^^^^^^^^^^
.. autofunction:: sympy.solvers.ode.ode_1st_linear

:obj:`Bernoulli`
^^^^^^^^^^^^^^^^
.. autofunction:: sympy.solvers.ode.ode_Bernoulli

:obj:`Liouville`
^^^^^^^^^^^^^^^^
.. autofunction:: sympy.solvers.ode.ode_Liouville

:obj:`Riccati_special_minus2`
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
.. autofunction:: sympy.solvers.ode.ode_Riccati_special_minus2

:obj:`nth_linear_constant_coeff_homogeneous`
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
.. autofunction:: sympy.solvers.ode.ode_nth_linear_constant_coeff_homogeneous

:obj:`nth_linear_constant_coeff_undetermined_coefficients`
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
.. autofunction:: sympy.solvers.ode.ode_nth_linear_constant_coeff_undetermined_coefficients

:obj:`nth_linear_constant_coeff_variation_of_parameters`
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
.. autofunction:: sympy.solvers.ode.ode_nth_linear_constant_coeff_variation_of_parameters

:obj:`separable`
^^^^^^^^^^^^^^^^
.. autofunction:: sympy.solvers.ode.ode_separable

:obj:`almost_linear`
^^^^^^^^^^^^^^^^^^^^
.. autofunction:: sympy.solvers.ode.ode_almost_linear

:obj:`linear_coefficients`
^^^^^^^^^^^^^^^^^^^^^^^^^^
.. autofunction:: sympy.solvers.ode.ode_linear_coefficients

:obj:`separable_reduced`
^^^^^^^^^^^^^^^^^^^^^^^^
.. autofunction:: sympy.solvers.ode.ode_separable_reduced

:obj:`lie_group`
^^^^^^^^^^^^^^^^
.. autofunction:: sympy.solvers.ode.ode_lie_group

:obj:`1st_power_series`
^^^^^^^^^^^^^^^^^^^^^^^
.. autofunction:: sympy.solvers.ode.ode_1st_power_series

:obj:`2nd_power_series_ordinary`
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
.. autofunction:: sympy.solvers.ode.ode_2nd_power_series_ordinary

:obj:`2nd_power_series_regular`
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
.. autofunction:: sympy.solvers.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.

:obj:`abaco1_simple`
^^^^^^^^^^^^^^^^^^^^
.. autofunction:: sympy.solvers.ode.lie_heuristic_abaco1_simple

:obj:`abaco1_product`
^^^^^^^^^^^^^^^^^^^^^
.. autofunction:: sympy.solvers.ode.lie_heuristic_abaco1_product

:obj:`bivariate`
^^^^^^^^^^^^^^^^
.. autofunction:: sympy.solvers.ode.lie_heuristic_bivariate

:obj:`chi`
^^^^^^^^^^
.. autofunction:: sympy.solvers.ode.lie_heuristic_chi

:obj:`abaco2_similar`
^^^^^^^^^^^^^^^^^^^^^
.. autofunction:: sympy.solvers.ode.lie_heuristic_abaco2_similar

:obj:`function_sum`
^^^^^^^^^^^^^^^^^^^
.. autofunction:: sympy.solvers.ode.lie_heuristic_function_sum

:obj:`abaco2_unique_unknown`
^^^^^^^^^^^^^^^^^^^^^^^^^^^^
.. autofunction:: sympy.solvers.ode.lie_heuristic_abaco2_unique_unknown

:obj:`abaco2_unique_general`
^^^^^^^^^^^^^^^^^^^^^^^^^^^^
.. autofunction:: sympy.solvers.ode.lie_heuristic_abaco2_unique_general

:obj:`linear`
^^^^^^^^^^^^^
.. autofunction:: sympy.solvers.ode.lie_heuristic_linear

Information on the ode module
-----------------------------

.. automodule:: sympy.solvers.ode