1.. _ode-docs: 2 3ODE 4=== 5 6.. module::sympy.solvers.ode 7 8.. automodule:: sympy.solvers.ode 9 10User Functions 11-------------- 12These are functions that are imported into the global namespace with ``from 13sympy import *``. These functions (unlike `Hint Functions`_, below) are 14intended for use by ordinary users of SymPy. 15 16dsolve 17^^^^^^ 18.. autofunction:: sympy.solvers.ode::dsolve 19 20dsolve_system 21^^^^^^^^^^^^^ 22.. autofunction:: sympy.solvers.ode.systems::dsolve_system 23 24classify_ode 25^^^^^^^^^^^^ 26.. autofunction:: sympy.solvers.ode::classify_ode 27 28checkodesol 29^^^^^^^^^^^ 30.. autofunction:: sympy.solvers.ode::checkodesol 31 32homogeneous_order 33^^^^^^^^^^^^^^^^^ 34.. autofunction:: sympy.solvers.ode::homogeneous_order 35 36infinitesimals 37^^^^^^^^^^^^^^ 38.. autofunction:: sympy.solvers.ode::infinitesimals 39 40checkinfsol 41^^^^^^^^^^^ 42.. autofunction:: sympy.solvers.ode::checkinfsol 43 44constantsimp 45^^^^^^^^^^^^ 46.. autofunction:: sympy.solvers.ode::constantsimp 47 48Hint Functions 49-------------- 50These functions are intended for internal use by 51:py:meth:`~sympy.solvers.ode.dsolve` and others. Unlike `User Functions`_, 52above, these are not intended for every-day use by ordinary SymPy users. 53Instead, functions such as :py:meth:`~sympy.solvers.ode.dsolve` should be used. 54Nonetheless, these functions contain useful information in their docstrings on 55the various ODE solving methods. For this reason, they are documented here. 56 57allhints 58^^^^^^^^ 59.. autodata:: sympy.solvers.ode::allhints 60 61odesimp 62^^^^^^^ 63.. autofunction:: sympy.solvers.ode.ode::odesimp 64 65constant_renumber 66^^^^^^^^^^^^^^^^^ 67.. autofunction:: sympy.solvers.ode.ode::constant_renumber 68 69sol_simplicity 70^^^^^^^^^^^^^^ 71.. autofunction:: sympy.solvers.ode.ode::ode_sol_simplicity 72 73factorable 74^^^^^^^^^^ 75.. autoclass:: sympy.solvers.ode.single::Factorable 76 :members: 77 781st_exact 79^^^^^^^^^ 80.. autoclass:: sympy.solvers.ode.single.FirstExact 81 :members: 82 831st_homogeneous_coeff_best 84^^^^^^^^^^^^^^^^^^^^^^^^^^ 85.. autoclass:: sympy.solvers.ode.single::HomogeneousCoeffBest 86 :members: 87 881st_homogeneous_coeff_subs_dep_div_indep 89^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ 90.. autoclass:: sympy.solvers.ode.single::HomogeneousCoeffSubsDepDivIndep 91 :members: 92 931st_homogeneous_coeff_subs_indep_div_dep 94^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ 95.. autoclass:: sympy.solvers.ode.single::HomogeneousCoeffSubsIndepDivDep 96 :members: 97 981st_linear 99^^^^^^^^^^ 100.. autoclass:: sympy.solvers.ode.single::FirstLinear 101 :members: 102 1031st_rational_riccati 104^^^^^^^^^^^^^^^^^^^^ 105.. autoclass:: sympy.solvers.ode.single::RationalRiccati 106 :members: 107 1082nd_linear_airy 109^^^^^^^^^^^^^^^ 110.. autoclass:: sympy.solvers.ode.single::SecondLinearAiry 111 :members: 112 1132nd_linear_bessel 114^^^^^^^^^^^^^^^^^ 115.. autoclass:: sympy.solvers.ode.single::SecondLinearBessel 116 :members: 117 118Bernoulli 119^^^^^^^^^ 120.. autoclass:: sympy.solvers.ode.single::Bernoulli 121 :members: 122 123Liouville 124^^^^^^^^^ 125.. autoclass:: sympy.solvers.ode.single::Liouville 126 :members: 127 128Riccati_special_minus2 129^^^^^^^^^^^^^^^^^^^^^^ 130.. autoclass:: sympy.solvers.ode.single::RiccatiSpecial 131 :members: 132 133nth_linear_constant_coeff_homogeneous 134^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ 135.. autoclass:: sympy.solvers.ode.single::NthLinearConstantCoeffHomogeneous 136 :members: 137 138nth_linear_constant_coeff_undetermined_coefficients 139^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ 140.. autoclass:: sympy.solvers.ode.single::NthLinearConstantCoeffUndeterminedCoefficients 141 :members: 142 143nth_linear_constant_coeff_variation_of_parameters 144^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ 145.. autoclass:: sympy.solvers.ode.single::NthLinearConstantCoeffVariationOfParameters 146 :members: 147 148nth_linear_euler_eq_homogeneous 149^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ 150.. autoclass:: sympy.solvers.ode.single::NthLinearEulerEqHomogeneous 151 :members: 152 153nth_linear_euler_eq_nonhomogeneous_variation_of_parameters 154^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ 155.. autoclass:: sympy.solvers.ode.single::NthLinearEulerEqNonhomogeneousVariationOfParameters 156 :members: 157 158nth_linear_euler_eq_nonhomogeneous_undetermined_coefficients 159^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ 160.. autoclass:: sympy.solvers.ode.single::NthLinearEulerEqNonhomogeneousUndeterminedCoefficients 161 :members: 162 163nth_algebraic 164^^^^^^^^^^^^^ 165.. autoclass:: sympy.solvers.ode.single::NthAlgebraic 166 :members: 167 168nth_order_reducible 169^^^^^^^^^^^^^^^^^^^ 170.. autoclass:: sympy.solvers.ode.single::NthOrderReducible 171 :members: 172 173separable 174^^^^^^^^^ 175.. autoclass:: sympy.solvers.ode.single::Separable 176 :members: 177 178almost_linear 179^^^^^^^^^^^^^ 180.. autoclass:: sympy.solvers.ode.single::AlmostLinear 181 :members: 182 183linear_coefficients 184^^^^^^^^^^^^^^^^^^^ 185.. autoclass:: sympy.solvers.ode.single::LinearCoefficients 186 :members: 187 188separable_reduced 189^^^^^^^^^^^^^^^^^ 190.. autoclass:: sympy.solvers.ode.single::SeparableReduced 191 :members: 192 193lie_group 194^^^^^^^^^ 195.. autoclass:: sympy.solvers.ode.single::LieGroup 196 :members: 197 1982nd_hypergeometric 199^^^^^^^^^^^^^^^^^^ 200.. autoclass:: sympy.solvers.ode.single::SecondHypergeometric 201 :members: 202 2031st_power_series 204^^^^^^^^^^^^^^^^ 205.. autofunction:: sympy.solvers.ode.ode::ode_1st_power_series 206 2072nd_power_series_ordinary 208^^^^^^^^^^^^^^^^^^^^^^^^^ 209.. autofunction:: sympy.solvers.ode.ode::ode_2nd_power_series_ordinary 210 2112nd_power_series_regular 212^^^^^^^^^^^^^^^^^^^^^^^^ 213.. autofunction:: sympy.solvers.ode.ode::ode_2nd_power_series_regular 214 215Lie heuristics 216-------------- 217These functions are intended for internal use of the Lie Group Solver. 218Nonetheless, they contain useful information in their docstrings on the algorithms 219implemented for the various heuristics. 220 221abaco1_simple 222^^^^^^^^^^^^^ 223.. autofunction:: sympy.solvers.ode.lie_group::lie_heuristic_abaco1_simple 224 225abaco1_product 226^^^^^^^^^^^^^^ 227.. autofunction:: sympy.solvers.ode.lie_group::lie_heuristic_abaco1_product 228 229bivariate 230^^^^^^^^^ 231.. autofunction:: sympy.solvers.ode.lie_group::lie_heuristic_bivariate 232 233chi 234^^^ 235.. autofunction:: sympy.solvers.ode.lie_group::lie_heuristic_chi 236 237abaco2_similar 238^^^^^^^^^^^^^^ 239.. autofunction:: sympy.solvers.ode.lie_group::lie_heuristic_abaco2_similar 240 241function_sum 242^^^^^^^^^^^^ 243.. autofunction:: sympy.solvers.ode.lie_group::lie_heuristic_function_sum 244 245abaco2_unique_unknown 246^^^^^^^^^^^^^^^^^^^^^ 247.. autofunction:: sympy.solvers.ode.lie_group::lie_heuristic_abaco2_unique_unknown 248 249abaco2_unique_general 250^^^^^^^^^^^^^^^^^^^^^ 251.. autofunction:: sympy.solvers.ode.lie_group::lie_heuristic_abaco2_unique_general 252 253linear 254^^^^^^ 255.. autofunction:: sympy.solvers.ode.lie_group::lie_heuristic_linear 256 257Rational Riccati Solver 258----------------------- 259These functions are intended for internal use to solve 260a first order Riccati differential equation with atleast 261one rational particular solution. 262 263riccati_normal 264^^^^^^^^^^^^^^ 265.. autofunction:: sympy.solvers.ode.riccati::riccati_normal 266 267riccati_inverse_normal 268^^^^^^^^^^^^^^^^^^^^^^ 269.. autofunction:: sympy.solvers.ode.riccati::riccati_inverse_normal 270 271riccati_reduced 272^^^^^^^^^^^^^^^ 273.. autofunction:: sympy.solvers.ode.riccati::riccati_reduced 274 275construct_c 276^^^^^^^^^^^ 277.. autofunction:: sympy.solvers.ode.riccati::construct_c 278 279construct_d 280^^^^^^^^^^^ 281.. autofunction:: sympy.solvers.ode.riccati::construct_d 282 283rational_laurent_series 284^^^^^^^^^^^^^^^^^^^^^^^ 285.. autofunction:: sympy.solvers.ode.riccati::rational_laurent_series 286 287compute_m_ybar 288^^^^^^^^^^^^^^ 289.. autofunction:: sympy.solvers.ode.riccati::compute_m_ybar 290 291solve_aux_eq 292^^^^^^^^^^^^ 293.. autofunction:: sympy.solvers.ode.riccati::solve_aux_eq 294 295remove_redundant_sols 296^^^^^^^^^^^^^^^^^^^^^ 297.. autofunction:: sympy.solvers.ode.riccati::remove_redundant_sols 298 299get_gen_sol_from_part_sol 300^^^^^^^^^^^^^^^^^^^^^^^^^ 301.. autofunction:: sympy.solvers.ode.riccati::get_gen_sol_from_part_sol 302 303solve_riccati 304^^^^^^^^^^^^^ 305.. autofunction:: sympy.solvers.ode.riccati::solve_riccati 306 307System of ODEs 308-------------- 309These functions are intended for internal use by 310:py:meth:`~sympy.solvers.ode.dsolve` for system of differential equations. 311 312Linear, 2 equations, Order 1, Type 6 313^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ 314.. autofunction:: sympy.solvers.ode.ode::_linear_2eq_order1_type6 315 316Linear, 2 equations, Order 1, Type 7 317^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ 318.. autofunction:: sympy.solvers.ode.ode::_linear_2eq_order1_type7 319 320Linear ODE to matrix 321^^^^^^^^^^^^^^^^^^^^ 322.. autofunction:: sympy.solvers.ode.systems::linear_ode_to_matrix 323 324Canonical Equations Converter 325^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ 326.. autofunction:: sympy.solvers.ode.systems::canonical_odes 327 328LinODESolve Systems Information 329^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ 330.. autofunction:: sympy.solvers.ode.systems::linodesolve_type 331 332Matrix Exponential Jordan Form 333^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ 334.. autofunction:: sympy.solvers.ode.systems::matrix_exp_jordan_form 335 336Matrix Exponential 337^^^^^^^^^^^^^^^^^^ 338.. autofunction:: sympy.solvers.ode.systems::matrix_exp 339 340Linear, n equations, Order 1 Solver 341^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ 342.. autofunction:: sympy.solvers.ode.systems::linodesolve 343 344Nonlinear, 2 equations, Order 1, Type 1 345^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ 346.. autofunction:: sympy.solvers.ode.ode::_nonlinear_2eq_order1_type1 347 348Nonlinear, 2 equations, Order 1, Type 2 349^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ 350.. autofunction:: sympy.solvers.ode.ode::_nonlinear_2eq_order1_type2 351 352Nonlinear, 2 equations, Order 1, Type 3 353^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ 354.. autofunction:: sympy.solvers.ode.ode::_nonlinear_2eq_order1_type3 355 356Nonlinear, 2 equations, Order 1, Type 4 357^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ 358.. autofunction:: sympy.solvers.ode.ode::_nonlinear_2eq_order1_type4 359 360Nonlinear, 2 equations, Order 1, Type 5 361^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ 362.. autofunction:: sympy.solvers.ode.ode::_nonlinear_2eq_order1_type5 363 364Nonlinear, 3 equations, Order 1, Type 1 365^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ 366.. autofunction:: sympy.solvers.ode.ode::_nonlinear_3eq_order1_type1 367 368Nonlinear, 3 equations, Order 1, Type 2 369^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ 370.. autofunction:: sympy.solvers.ode.ode::_nonlinear_3eq_order1_type2 371 372Nonlinear, 3 equations, Order 1, Type 3 373^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ 374.. autofunction:: sympy.solvers.ode.ode::_nonlinear_3eq_order1_type3 375 376Nonlinear, 3 equations, Order 1, Type 4 377^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ 378.. autofunction:: sympy.solvers.ode.ode::_nonlinear_3eq_order1_type4 379 380Nonlinear, 3 equations, Order 1, Type 5 381^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ 382.. autofunction:: sympy.solvers.ode.ode::_nonlinear_3eq_order1_type5 383 384Information on the ode module 385----------------------------- 386 387.. automodule:: sympy.solvers.ode.ode 388 389Internal functions 390^^^^^^^^^^^^^^^^^^ 391 392These functions are not intended for end-user use. 393 394.. autofunction:: sympy.solvers.ode.ode::_handle_Integral 395