.. _ode-docs: 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. dsolve ^^^^^^ .. autofunction:: sympy.solvers.ode::dsolve dsolve_system ^^^^^^^^^^^^^ .. autofunction:: sympy.solvers.ode.systems::dsolve_system classify_ode ^^^^^^^^^^^^ .. autofunction:: sympy.solvers.ode::classify_ode checkodesol ^^^^^^^^^^^ .. autofunction:: sympy.solvers.ode::checkodesol homogeneous_order ^^^^^^^^^^^^^^^^^ .. autofunction:: sympy.solvers.ode::homogeneous_order infinitesimals ^^^^^^^^^^^^^^ .. autofunction:: sympy.solvers.ode::infinitesimals checkinfsol ^^^^^^^^^^^ .. autofunction:: sympy.solvers.ode::checkinfsol constantsimp ^^^^^^^^^^^^ .. autofunction:: sympy.solvers.ode::constantsimp 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. allhints ^^^^^^^^ .. autodata:: sympy.solvers.ode::allhints odesimp ^^^^^^^ .. autofunction:: sympy.solvers.ode.ode::odesimp constant_renumber ^^^^^^^^^^^^^^^^^ .. autofunction:: sympy.solvers.ode.ode::constant_renumber sol_simplicity ^^^^^^^^^^^^^^ .. autofunction:: sympy.solvers.ode.ode::ode_sol_simplicity factorable ^^^^^^^^^^ .. autoclass:: sympy.solvers.ode.single::Factorable 1st_exact ^^^^^^^^^ .. autofunction:: sympy.solvers.ode.ode::ode_1st_exact 1st_homogeneous_coeff_best ^^^^^^^^^^^^^^^^^^^^^^^^^^ .. autofunction:: sympy.solvers.ode.ode::ode_1st_homogeneous_coeff_best 1st_homogeneous_coeff_subs_dep_div_indep ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ .. autofunction:: sympy.solvers.ode.ode::ode_1st_homogeneous_coeff_subs_dep_div_indep 1st_homogeneous_coeff_subs_indep_div_dep ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ .. autofunction:: sympy.solvers.ode.ode::ode_1st_homogeneous_coeff_subs_indep_div_dep 1st_linear ^^^^^^^^^^ .. autoclass:: sympy.solvers.ode.single::FirstLinear 2nd_linear_airy ^^^^^^^^^^^^^^^ .. autofunction:: sympy.solvers.ode.ode::ode_2nd_linear_airy 2nd_linear_bessel ^^^^^^^^^^^^^^^^^ .. autofunction:: sympy.solvers.ode.ode::ode_2nd_linear_bessel Bernoulli ^^^^^^^^^ .. autoclass:: sympy.solvers.ode.single::Bernoulli Liouville ^^^^^^^^^ .. autofunction:: sympy.solvers.ode.ode::ode_Liouville Riccati_special_minus2 ^^^^^^^^^^^^^^^^^^^^^^ .. autoclass:: sympy.solvers.ode.single::RiccatiSpecial nth_linear_constant_coeff_homogeneous ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ .. autofunction:: sympy.solvers.ode.ode::ode_nth_linear_constant_coeff_homogeneous nth_linear_constant_coeff_undetermined_coefficients ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ .. autofunction:: sympy.solvers.ode.ode::ode_nth_linear_constant_coeff_undetermined_coefficients nth_linear_constant_coeff_variation_of_parameters ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ .. autofunction:: sympy.solvers.ode.ode::ode_nth_linear_constant_coeff_variation_of_parameters nth_algebraic ^^^^^^^^^^^^^ .. autoclass:: sympy.solvers.ode.single::NthAlgebraic nth_order_reducible ^^^^^^^^^^^^^^^^^^^ .. autofunction:: sympy.solvers.ode.ode::ode_nth_order_reducible separable ^^^^^^^^^ .. autofunction:: sympy.solvers.ode.ode::ode_separable almost_linear ^^^^^^^^^^^^^ .. autoclass:: sympy.solvers.ode.single::AlmostLinear linear_coefficients ^^^^^^^^^^^^^^^^^^^ .. autofunction:: sympy.solvers.ode.ode::ode_linear_coefficients separable_reduced ^^^^^^^^^^^^^^^^^ .. autofunction:: sympy.solvers.ode.ode::ode_separable_reduced lie_group ^^^^^^^^^ .. autofunction:: sympy.solvers.ode.ode::ode_lie_group 1st_power_series ^^^^^^^^^^^^^^^^ .. autofunction:: sympy.solvers.ode.ode::ode_1st_power_series 2nd_power_series_ordinary ^^^^^^^^^^^^^^^^^^^^^^^^^ .. autofunction:: sympy.solvers.ode.ode::ode_2nd_power_series_ordinary 2nd_power_series_regular ^^^^^^^^^^^^^^^^^^^^^^^^ .. 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. abaco1_simple ^^^^^^^^^^^^^ .. autofunction:: sympy.solvers.ode.ode::lie_heuristic_abaco1_simple abaco1_product ^^^^^^^^^^^^^^ .. autofunction:: sympy.solvers.ode.ode::lie_heuristic_abaco1_product bivariate ^^^^^^^^^ .. autofunction:: sympy.solvers.ode.ode::lie_heuristic_bivariate chi ^^^ .. autofunction:: sympy.solvers.ode.ode::lie_heuristic_chi abaco2_similar ^^^^^^^^^^^^^^ .. autofunction:: sympy.solvers.ode.ode::lie_heuristic_abaco2_similar function_sum ^^^^^^^^^^^^ .. autofunction:: sympy.solvers.ode.ode::lie_heuristic_function_sum abaco2_unique_unknown ^^^^^^^^^^^^^^^^^^^^^ .. autofunction:: sympy.solvers.ode.ode::lie_heuristic_abaco2_unique_unknown abaco2_unique_general ^^^^^^^^^^^^^^^^^^^^^ .. autofunction:: sympy.solvers.ode.ode::lie_heuristic_abaco2_unique_general linear ^^^^^^ .. autofunction:: sympy.solvers.ode.ode::lie_heuristic_linear System of ODEs -------------- These functions are intended for internal use by :py:meth:`~sympy.solvers.ode.dsolve` for system of differential equations. Linear, 2 equations, Order 1, Type 6 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ .. autofunction:: sympy.solvers.ode.ode::_linear_2eq_order1_type6 Linear, 2 equations, Order 1, Type 7 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ .. autofunction:: sympy.solvers.ode.ode::_linear_2eq_order1_type7 Linear ODE to matrix ^^^^^^^^^^^^^^^^^^^^ .. autofunction:: sympy.solvers.ode.systems::linear_ode_to_matrix Canonical Equations Converter ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ .. autofunction:: sympy.solvers.ode.systems::canonical_odes LinODESolve Systems Information ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ .. autofunction:: sympy.solvers.ode.systems::linodesolve_type Matrix Exponential Jordan Form ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ .. autofunction:: sympy.solvers.ode.systems::matrix_exp_jordan_form Matrix Exponential ^^^^^^^^^^^^^^^^^^ .. autofunction:: sympy.solvers.ode.systems::matrix_exp Linear, n equations, Order 1 Solver ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ .. autofunction:: sympy.solvers.ode.systems::linodesolve Nonlinear, 2 equations, Order 1, Type 1 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ .. autofunction:: sympy.solvers.ode.ode::_nonlinear_2eq_order1_type1 Nonlinear, 2 equations, Order 1, Type 2 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ .. autofunction:: sympy.solvers.ode.ode::_nonlinear_2eq_order1_type2 Nonlinear, 2 equations, Order 1, Type 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ .. autofunction:: sympy.solvers.ode.ode::_nonlinear_2eq_order1_type3 Nonlinear, 2 equations, Order 1, Type 4 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ .. autofunction:: sympy.solvers.ode.ode::_nonlinear_2eq_order1_type4 Nonlinear, 2 equations, Order 1, Type 5 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ .. autofunction:: sympy.solvers.ode.ode::_nonlinear_2eq_order1_type5 Nonlinear, 3 equations, Order 1, Type 1 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ .. autofunction:: sympy.solvers.ode.ode::_nonlinear_3eq_order1_type1 Nonlinear, 3 equations, Order 1, Type 2 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ .. autofunction:: sympy.solvers.ode.ode::_nonlinear_3eq_order1_type2 Nonlinear, 3 equations, Order 1, Type 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ .. autofunction:: sympy.solvers.ode.ode::_nonlinear_3eq_order1_type3 Nonlinear, 3 equations, Order 1, Type 4 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ .. autofunction:: sympy.solvers.ode.ode::_nonlinear_3eq_order1_type4 Nonlinear, 3 equations, Order 1, Type 5 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ .. autofunction:: sympy.solvers.ode.ode::_nonlinear_3eq_order1_type5 Information on the ode module ----------------------------- .. automodule:: sympy.solvers.ode.ode Internal functions ^^^^^^^^^^^^^^^^^^ These functions are not intended for end-user use. .. autofunction:: sympy.solvers.ode.ode::_undetermined_coefficients_match .. autofunction:: sympy.solvers.ode.ode::_handle_Integral