# # This is the module for ODE solver classes for single ODEs. # from __future__ import annotations from typing import ClassVar, Iterator from .riccati import match_riccati, solve_riccati from sympy.core import Add, S, Pow, Rational from sympy.core.cache import cached_property from sympy.core.exprtools import factor_terms from sympy.core.expr import Expr from sympy.core.function import AppliedUndef, Derivative, diff, Function, expand, Subs, _mexpand from sympy.core.numbers import zoo from sympy.core.relational import Equality, Eq from sympy.core.symbol import Symbol, Dummy, Wild from sympy.core.mul import Mul from sympy.functions import exp, tan, log, sqrt, besselj, bessely, cbrt, airyai, airybi from sympy.integrals import Integral from sympy.polys import Poly from sympy.polys.polytools import cancel, factor, degree from sympy.simplify import collect, simplify, separatevars, logcombine, posify # type: ignore from sympy.simplify.radsimp import fraction from sympy.utilities import numbered_symbols from sympy.solvers.solvers import solve from sympy.solvers.deutils import ode_order, _preprocess from sympy.polys.matrices.linsolve import _lin_eq2dict from sympy.polys.solvers import PolyNonlinearError from .hypergeometric import equivalence_hypergeometric, match_2nd_2F1_hypergeometric, \ get_sol_2F1_hypergeometric, match_2nd_hypergeometric from .nonhomogeneous import _get_euler_characteristic_eq_sols, _get_const_characteristic_eq_sols, \ _solve_undetermined_coefficients, _solve_variation_of_parameters, _test_term, _undetermined_coefficients_match, \ _get_simplified_sol from .lie_group import _ode_lie_group class ODEMatchError(NotImplementedError): """Raised if a SingleODESolver is asked to solve an ODE it does not match""" pass class SingleODEProblem: """Represents an ordinary differential equation (ODE) This class is used internally in the by dsolve and related functions/classes so that properties of an ODE can be computed efficiently. Examples ======== This class is used internally by dsolve. To instantiate an instance directly first define an ODE problem: >>> from sympy import Function, Symbol >>> x = Symbol('x') >>> f = Function('f') >>> eq = f(x).diff(x, 2) Now you can create a SingleODEProblem instance and query its properties: >>> from sympy.solvers.ode.single import SingleODEProblem >>> problem = SingleODEProblem(f(x).diff(x), f(x), x) >>> problem.eq Derivative(f(x), x) >>> problem.func f(x) >>> problem.sym x """ # Instance attributes: eq = None # type: Expr func = None # type: AppliedUndef sym = None # type: Symbol _order = None # type: int _eq_expanded = None # type: Expr _eq_preprocessed = None # type: Expr _eq_high_order_free = None def __init__(self, eq, func, sym, prep=True, **kwargs): assert isinstance(eq, Expr) assert isinstance(func, AppliedUndef) assert isinstance(sym, Symbol) assert isinstance(prep, bool) self.eq = eq self.func = func self.sym = sym self.prep = prep self.params = kwargs @cached_property def order(self) -> int: return ode_order(self.eq, self.func) @cached_property def eq_preprocessed(self) -> Expr: return self._get_eq_preprocessed() @cached_property def eq_high_order_free(self) -> Expr: a = Wild('a', exclude=[self.func]) c1 = Wild('c1', exclude=[self.sym]) # Precondition to try remove f(x) from highest order derivative reduced_eq = None if self.eq.is_Add: deriv_coef = self.eq.coeff(self.func.diff(self.sym, self.order)) if deriv_coef not in (1, 0): r = deriv_coef.match(a*self.func**c1) if r and r[c1]: den = self.func**r[c1] reduced_eq = Add(*[arg/den for arg in self.eq.args]) if not reduced_eq: reduced_eq = expand(self.eq) return reduced_eq @cached_property def eq_expanded(self) -> Expr: return expand(self.eq_preprocessed) def _get_eq_preprocessed(self) -> Expr: if self.prep: process_eq, process_func = _preprocess(self.eq, self.func) if process_func != self.func: raise ValueError else: process_eq = self.eq return process_eq def get_numbered_constants(self, num=1, start=1, prefix='C') -> list[Symbol]: """ Returns a list of constants that do not occur in eq already. """ ncs = self.iter_numbered_constants(start, prefix) Cs = [next(ncs) for i in range(num)] return Cs def iter_numbered_constants(self, start=1, prefix='C') -> Iterator[Symbol]: """ Returns an iterator of constants that do not occur in eq already. """ atom_set = self.eq.free_symbols func_set = self.eq.atoms(Function) if func_set: atom_set |= {Symbol(str(f.func)) for f in func_set} return numbered_symbols(start=start, prefix=prefix, exclude=atom_set) @cached_property def is_autonomous(self): u = Dummy('u') x = self.sym syms = self.eq.subs(self.func, u).free_symbols return x not in syms def get_linear_coefficients(self, eq, func, order): r""" Matches a differential equation to the linear form: .. math:: a_n(x) y^{(n)} + \cdots + a_1(x)y' + a_0(x) y + B(x) = 0 Returns a dict of order:coeff terms, where order is the order of the derivative on each term, and coeff is the coefficient of that derivative. The key ``-1`` holds the function `B(x)`. Returns ``None`` if the ODE is not linear. This function assumes that ``func`` has already been checked to be good. Examples ======== >>> from sympy import Function, cos, sin >>> from sympy.abc import x >>> from sympy.solvers.ode.single import SingleODEProblem >>> f = Function('f') >>> eq = f(x).diff(x, 3) + 2*f(x).diff(x) + \ ... x*f(x).diff(x, 2) + cos(x)*f(x).diff(x) + x - f(x) - \ ... sin(x) >>> obj = SingleODEProblem(eq, f(x), x) >>> obj.get_linear_coefficients(eq, f(x), 3) {-1: x - sin(x), 0: -1, 1: cos(x) + 2, 2: x, 3: 1} >>> eq = f(x).diff(x, 3) + 2*f(x).diff(x) + \ ... x*f(x).diff(x, 2) + cos(x)*f(x).diff(x) + x - f(x) - \ ... sin(f(x)) >>> obj = SingleODEProblem(eq, f(x), x) >>> obj.get_linear_coefficients(eq, f(x), 3) == None True """ f = func.func x = func.args[0] symset = {Derivative(f(x), x, i) for i in range(order+1)} try: rhs, lhs_terms = _lin_eq2dict(eq, symset) except PolyNonlinearError: return None if rhs.has(func) or any(c.has(func) for c in lhs_terms.values()): return None terms = {i: lhs_terms.get(f(x).diff(x, i), S.Zero) for i in range(order+1)} terms[-1] = rhs return terms # TODO: Add methods that can be used by many ODE solvers: # order # is_linear() # get_linear_coefficients() # eq_prepared (the ODE in prepared form) class SingleODESolver: """ Base class for Single ODE solvers. Subclasses should implement the _matches and _get_general_solution methods. This class is not intended to be instantiated directly but its subclasses are as part of dsolve. Examples ======== You can use a subclass of SingleODEProblem to solve a particular type of ODE. We first define a particular ODE problem: >>> from sympy import Function, Symbol >>> x = Symbol('x') >>> f = Function('f') >>> eq = f(x).diff(x, 2) Now we solve this problem using the NthAlgebraic solver which is a subclass of SingleODESolver: >>> from sympy.solvers.ode.single import NthAlgebraic, SingleODEProblem >>> problem = SingleODEProblem(eq, f(x), x) >>> solver = NthAlgebraic(problem) >>> solver.get_general_solution() [Eq(f(x), _C*x + _C)] The normal way to solve an ODE is to use dsolve (which would use NthAlgebraic and other solvers internally). When using dsolve a number of other things are done such as evaluating integrals, simplifying the solution and renumbering the constants: >>> from sympy import dsolve >>> dsolve(eq, hint='nth_algebraic') Eq(f(x), C1 + C2*x) """ # Subclasses should store the hint name (the argument to dsolve) in this # attribute hint: ClassVar[str] # Subclasses should define this to indicate if they support an _Integral # hint. has_integral: ClassVar[bool] # The ODE to be solved ode_problem = None # type: SingleODEProblem # Cache whether or not the equation has matched the method _matched: bool | None = None # Subclasses should store in this attribute the list of order(s) of ODE # that subclass can solve or leave it to None if not specific to any order order: list | None = None def __init__(self, ode_problem): self.ode_problem = ode_problem def matches(self) -> bool: if self.order is not None and self.ode_problem.order not in self.order: self._matched = False return self._matched if self._matched is None: self._matched = self._matches() return self._matched def get_general_solution(self, *, simplify: bool = True) -> list[Equality]: if not self.matches(): msg = "%s solver cannot solve:\n%s" raise ODEMatchError(msg % (self.hint, self.ode_problem.eq)) return self._get_general_solution(simplify_flag=simplify) def _matches(self) -> bool: msg = "Subclasses of SingleODESolver should implement matches." raise NotImplementedError(msg) def _get_general_solution(self, *, simplify_flag: bool = True) -> list[Equality]: msg = "Subclasses of SingleODESolver should implement get_general_solution." raise NotImplementedError(msg) class SinglePatternODESolver(SingleODESolver): '''Superclass for ODE solvers based on pattern matching''' def wilds(self): prob = self.ode_problem f = prob.func.func x = prob.sym order = prob.order return self._wilds(f, x, order) def wilds_match(self): match = self._wilds_match return [match.get(w, S.Zero) for w in self.wilds()] def _matches(self): eq = self.ode_problem.eq_expanded f = self.ode_problem.func.func x = self.ode_problem.sym order = self.ode_problem.order df = f(x).diff(x, order) if order not in [1, 2]: return False pattern = self._equation(f(x), x, order) if not pattern.coeff(df).has(Wild): eq = expand(eq / eq.coeff(df)) eq = eq.collect([f(x).diff(x), f(x)], func = cancel) self._wilds_match = match = eq.match(pattern) if match is not None: return self._verify(f(x)) return False def _verify(self, fx) -> bool: return True def _wilds(self, f, x, order): msg = "Subclasses of SingleODESolver should implement _wilds" raise NotImplementedError(msg) def _equation(self, fx, x, order): msg = "Subclasses of SingleODESolver should implement _equation" raise NotImplementedError(msg) class NthAlgebraic(SingleODESolver): r""" Solves an `n`\th order ordinary differential equation using algebra and integrals. There is no general form for the kind of equation that this can solve. The the equation is solved algebraically treating differentiation as an invertible algebraic function. Examples ======== >>> from sympy import Function, dsolve, Eq >>> from sympy.abc import x >>> f = Function('f') >>> eq = Eq(f(x) * (f(x).diff(x)**2 - 1), 0) >>> dsolve(eq, f(x), hint='nth_algebraic') [Eq(f(x), 0), Eq(f(x), C1 - x), Eq(f(x), C1 + x)] Note that this solver can return algebraic solutions that do not have any integration constants (f(x) = 0 in the above example). """ hint = 'nth_algebraic' has_integral = True # nth_algebraic_Integral hint def _matches(self): r""" Matches any differential equation that nth_algebraic can solve. Uses `sympy.solve` but teaches it how to integrate derivatives. This involves calling `sympy.solve` and does most of the work of finding a solution (apart from evaluating the integrals). """ eq = self.ode_problem.eq func = self.ode_problem.func var = self.ode_problem.sym # Derivative that solve can handle: diffx = self._get_diffx(var) # Replace derivatives wrt the independent variable with diffx def replace(eq, var): def expand_diffx(*args): differand, diffs = args[0], args[1:] toreplace = differand for v, n in diffs: for _ in range(n): if v == var: toreplace = diffx(toreplace) else: toreplace = Derivative(toreplace, v) return toreplace return eq.replace(Derivative, expand_diffx) # Restore derivatives in solution afterwards def unreplace(eq, var): return eq.replace(diffx, lambda e: Derivative(e, var)) subs_eqn = replace(eq, var) try: # turn off simplification to protect Integrals that have # _t instead of fx in them and would otherwise factor # as t_*Integral(1, x) solns = solve(subs_eqn, func, simplify=False) except NotImplementedError: solns = [] solns = [simplify(unreplace(soln, var)) for soln in solns] solns = [Equality(func, soln) for soln in solns] self.solutions = solns return len(solns) != 0 def _get_general_solution(self, *, simplify_flag: bool = True): return self.solutions # This needs to produce an invertible function but the inverse depends # which variable we are integrating with respect to. Since the class can # be stored in cached results we need to ensure that we always get the # same class back for each particular integration variable so we store these # classes in a global dict: _diffx_stored: dict[Symbol, type[Function]] = {} @staticmethod def _get_diffx(var): diffcls = NthAlgebraic._diffx_stored.get(var, None) if diffcls is None: # A class that behaves like Derivative wrt var but is "invertible". class diffx(Function): def inverse(self): # don't use integrate here because fx has been replaced by _t # in the equation; integrals will not be correct while solve # is at work. return lambda expr: Integral(expr, var) + Dummy('C') diffcls = NthAlgebraic._diffx_stored.setdefault(var, diffx) return diffcls class FirstExact(SinglePatternODESolver): r""" Solves 1st order exact ordinary differential equations. A 1st order differential equation is called exact if it is the total differential of a function. That is, the differential equation .. math:: P(x, y) \,\partial{}x + Q(x, y) \,\partial{}y = 0 is exact if there is some function `F(x, y)` such that `P(x, y) = \partial{}F/\partial{}x` and `Q(x, y) = \partial{}F/\partial{}y`. It can be shown that a necessary and sufficient condition for a first order ODE to be exact is that `\partial{}P/\partial{}y = \partial{}Q/\partial{}x`. Then, the solution will be as given below:: >>> from sympy import Function, Eq, Integral, symbols, pprint >>> x, y, t, x0, y0, C1= symbols('x,y,t,x0,y0,C1') >>> P, Q, F= map(Function, ['P', 'Q', 'F']) >>> pprint(Eq(Eq(F(x, y), Integral(P(t, y), (t, x0, x)) + ... Integral(Q(x0, t), (t, y0, y))), C1)) x y / / | | F(x, y) = | P(t, y) dt + | Q(x0, t) dt = C1 | | / / x0 y0 Where the first partials of `P` and `Q` exist and are continuous in a simply connected region. A note: SymPy currently has no way to represent inert substitution on an expression, so the hint ``1st_exact_Integral`` will return an integral with `dy`. This is supposed to represent the function that you are solving for. Examples ======== >>> from sympy import Function, dsolve, cos, sin >>> from sympy.abc import x >>> f = Function('f') >>> dsolve(cos(f(x)) - (x*sin(f(x)) - f(x)**2)*f(x).diff(x), ... f(x), hint='1st_exact') Eq(x*cos(f(x)) + f(x)**3/3, C1) References ========== - https://en.wikipedia.org/wiki/Exact_differential_equation - M. Tenenbaum & H. Pollard, "Ordinary Differential Equations", Dover 1963, pp. 73 # indirect doctest """ hint = "1st_exact" has_integral = True order = [1] def _wilds(self, f, x, order): P = Wild('P', exclude=[f(x).diff(x)]) Q = Wild('Q', exclude=[f(x).diff(x)]) return P, Q def _equation(self, fx, x, order): P, Q = self.wilds() return P + Q*fx.diff(x) def _verify(self, fx) -> bool: P, Q = self.wilds() x = self.ode_problem.sym y = Dummy('y') m, n = self.wilds_match() m = m.subs(fx, y) n = n.subs(fx, y) numerator = cancel(m.diff(y) - n.diff(x)) if numerator.is_zero: # Is exact return True else: # The following few conditions try to convert a non-exact # differential equation into an exact one. # References: # 1. Differential equations with applications # and historical notes - George E. Simmons # 2. https://math.okstate.edu/people/binegar/2233-S99/2233-l12.pdf factor_n = cancel(numerator/n) factor_m = cancel(-numerator/m) if y not in factor_n.free_symbols: # If (dP/dy - dQ/dx) / Q = f(x) # then exp(integral(f(x))*equation becomes exact factor = factor_n integration_variable = x elif x not in factor_m.free_symbols: # If (dP/dy - dQ/dx) / -P = f(y) # then exp(integral(f(y))*equation becomes exact factor = factor_m integration_variable = y else: # Couldn't convert to exact return False factor = exp(Integral(factor, integration_variable)) m *= factor n *= factor self._wilds_match[P] = m.subs(y, fx) self._wilds_match[Q] = n.subs(y, fx) return True def _get_general_solution(self, *, simplify_flag: bool = True): m, n = self.wilds_match() fx = self.ode_problem.func x = self.ode_problem.sym (C1,) = self.ode_problem.get_numbered_constants(num=1) y = Dummy('y') m = m.subs(fx, y) n = n.subs(fx, y) gen_sol = Eq(Subs(Integral(m, x) + Integral(n - Integral(m, x).diff(y), y), y, fx), C1) return [gen_sol] class FirstLinear(SinglePatternODESolver): r""" Solves 1st order linear differential equations. These are differential equations of the form .. math:: dy/dx + P(x) y = Q(x)\text{.} These kinds of differential equations can be solved in a general way. The integrating factor `e^{\int P(x) \,dx}` will turn the equation into a separable equation. The general solution is:: >>> from sympy import Function, dsolve, Eq, pprint, diff, sin >>> from sympy.abc import x >>> f, P, Q = map(Function, ['f', 'P', 'Q']) >>> genform = Eq(f(x).diff(x) + P(x)*f(x), Q(x)) >>> pprint(genform) d P(x)*f(x) + --(f(x)) = Q(x) dx >>> pprint(dsolve(genform, f(x), hint='1st_linear_Integral')) / / \ | | | | | / | / | | | | | | | | P(x) dx | - | P(x) dx | | | | | | | / | / f(x) = |C1 + | Q(x)*e dx|*e | | | \ / / Examples ======== >>> f = Function('f') >>> pprint(dsolve(Eq(x*diff(f(x), x) - f(x), x**2*sin(x)), ... f(x), '1st_linear')) f(x) = x*(C1 - cos(x)) References ========== - https://en.wikipedia.org/wiki/Linear_differential_equation#First-order_equation_with_variable_coefficients - M. Tenenbaum & H. Pollard, "Ordinary Differential Equations", Dover 1963, pp. 92 # indirect doctest """ hint = '1st_linear' has_integral = True order = [1] def _wilds(self, f, x, order): P = Wild('P', exclude=[f(x)]) Q = Wild('Q', exclude=[f(x), f(x).diff(x)]) return P, Q def _equation(self, fx, x, order): P, Q = self.wilds() return fx.diff(x) + P*fx - Q def _get_general_solution(self, *, simplify_flag: bool = True): P, Q = self.wilds_match() fx = self.ode_problem.func x = self.ode_problem.sym (C1,) = self.ode_problem.get_numbered_constants(num=1) gensol = Eq(fx, ((C1 + Integral(Q*exp(Integral(P, x)), x)) * exp(-Integral(P, x)))) return [gensol] class AlmostLinear(SinglePatternODESolver): r""" Solves an almost-linear differential equation. The general form of an almost linear differential equation is .. math:: a(x) g'(f(x)) f'(x) + b(x) g(f(x)) + c(x) Here `f(x)` is the function to be solved for (the dependent variable). The substitution `g(f(x)) = u(x)` leads to a linear differential equation for `u(x)` of the form `a(x) u' + b(x) u + c(x) = 0`. This can be solved for `u(x)` by the `first_linear` hint and then `f(x)` is found by solving `g(f(x)) = u(x)`. See Also ======== :obj:`sympy.solvers.ode.single.FirstLinear` Examples ======== >>> from sympy import dsolve, Function, pprint, sin, cos >>> from sympy.abc import x >>> f = Function('f') >>> d = f(x).diff(x) >>> eq = x*d + x*f(x) + 1 >>> dsolve(eq, f(x), hint='almost_linear') Eq(f(x), (C1 - Ei(x))*exp(-x)) >>> pprint(dsolve(eq, f(x), hint='almost_linear')) -x f(x) = (C1 - Ei(x))*e >>> example = cos(f(x))*f(x).diff(x) + sin(f(x)) + 1 >>> pprint(example) d sin(f(x)) + cos(f(x))*--(f(x)) + 1 dx >>> pprint(dsolve(example, f(x), hint='almost_linear')) / -x \ / -x \ [f(x) = pi - asin\C1*e - 1/, f(x) = asin\C1*e - 1/] References ========== - Joel Moses, "Symbolic Integration - The Stormy Decade", Communications of the ACM, Volume 14, Number 8, August 1971, pp. 558 """ hint = "almost_linear" has_integral = True order = [1] def _wilds(self, f, x, order): P = Wild('P', exclude=[f(x).diff(x)]) Q = Wild('Q', exclude=[f(x).diff(x)]) return P, Q def _equation(self, fx, x, order): P, Q = self.wilds() return P*fx.diff(x) + Q def _verify(self, fx): a, b = self.wilds_match() c, b = b.as_independent(fx) if b.is_Add else (S.Zero, b) # a, b and c are the function a(x), b(x) and c(x) respectively. # c(x) is obtained by separating out b as terms with and without fx i.e, l(y) # The following conditions checks if the given equation is an almost-linear differential equation using the fact that # a(x)*(l(y))' / l(y)' is independent of l(y) if b.diff(fx) != 0 and not simplify(b.diff(fx)/a).has(fx): self.ly = factor_terms(b).as_independent(fx, as_Add=False)[1] # Gives the term containing fx i.e., l(y) self.ax = a / self.ly.diff(fx) self.cx = -c # cx is taken as -c(x) to simplify expression in the solution integral self.bx = factor_terms(b) / self.ly return True return False def _get_general_solution(self, *, simplify_flag: bool = True): x = self.ode_problem.sym (C1,) = self.ode_problem.get_numbered_constants(num=1) gensol = Eq(self.ly, ((C1 + Integral((self.cx/self.ax)*exp(Integral(self.bx/self.ax, x)), x)) * exp(-Integral(self.bx/self.ax, x)))) return [gensol] class Bernoulli(SinglePatternODESolver): r""" Solves Bernoulli differential equations. These are equations of the form .. math:: dy/dx + P(x) y = Q(x) y^n\text{, }n \ne 1`\text{.} The substitution `w = 1/y^{1-n}` will transform an equation of this form into one that is linear (see the docstring of :obj:`~sympy.solvers.ode.single.FirstLinear`). The general solution is:: >>> from sympy import Function, dsolve, Eq, pprint >>> from sympy.abc import x, n >>> f, P, Q = map(Function, ['f', 'P', 'Q']) >>> genform = Eq(f(x).diff(x) + P(x)*f(x), Q(x)*f(x)**n) >>> pprint(genform) d n P(x)*f(x) + --(f(x)) = Q(x)*f (x) dx >>> pprint(dsolve(genform, f(x), hint='Bernoulli_Integral'), num_columns=110) -1 ----- n - 1 // / / \ \ || | | | | || | / | / | / | || | | | | | | | || | -(n - 1)* | P(x) dx | -(n - 1)* | P(x) dx | (n - 1)* | P(x) dx| || | | | | | | | || | / | / | / | f(x) = ||C1 - n* | Q(x)*e dx + | Q(x)*e dx|*e | || | | | | \\ / / / / Note that the equation is separable when `n = 1` (see the docstring of :obj:`~sympy.solvers.ode.single.Separable`). >>> pprint(dsolve(Eq(f(x).diff(x) + P(x)*f(x), Q(x)*f(x)), f(x), ... hint='separable_Integral')) f(x) / | / | 1 | | - dy = C1 + | (-P(x) + Q(x)) dx | y | | / / Examples ======== >>> from sympy import Function, dsolve, Eq, pprint, log >>> from sympy.abc import x >>> f = Function('f') >>> pprint(dsolve(Eq(x*f(x).diff(x) + f(x), log(x)*f(x)**2), ... f(x), hint='Bernoulli')) 1 f(x) = ----------------- C1*x + log(x) + 1 References ========== - https://en.wikipedia.org/wiki/Bernoulli_differential_equation - M. Tenenbaum & H. Pollard, "Ordinary Differential Equations", Dover 1963, pp. 95 # indirect doctest """ hint = "Bernoulli" has_integral = True order = [1] def _wilds(self, f, x, order): P = Wild('P', exclude=[f(x)]) Q = Wild('Q', exclude=[f(x)]) n = Wild('n', exclude=[x, f(x), f(x).diff(x)]) return P, Q, n def _equation(self, fx, x, order): P, Q, n = self.wilds() return fx.diff(x) + P*fx - Q*fx**n def _get_general_solution(self, *, simplify_flag: bool = True): P, Q, n = self.wilds_match() fx = self.ode_problem.func x = self.ode_problem.sym (C1,) = self.ode_problem.get_numbered_constants(num=1) if n==1: gensol = Eq(log(fx), ( C1 + Integral((-P + Q), x) )) else: gensol = Eq(fx**(1-n), ( (C1 - (n - 1) * Integral(Q*exp(-n*Integral(P, x)) * exp(Integral(P, x)), x) ) * exp(-(1 - n)*Integral(P, x))) ) return [gensol] class Factorable(SingleODESolver): r""" Solves equations having a solvable factor. This function is used to solve the equation having factors. Factors may be of type algebraic or ode. It will try to solve each factor independently. Factors will be solved by calling dsolve. We will return the list of solutions. Examples ======== >>> from sympy import Function, dsolve, pprint >>> from sympy.abc import x >>> f = Function('f') >>> eq = (f(x)**2-4)*(f(x).diff(x)+f(x)) >>> pprint(dsolve(eq, f(x))) -x [f(x) = 2, f(x) = -2, f(x) = C1*e ] """ hint = "factorable" has_integral = False def _matches(self): eq_orig = self.ode_problem.eq f = self.ode_problem.func.func x = self.ode_problem.sym df = f(x).diff(x) self.eqs = [] eq = eq_orig.collect(f(x), func = cancel) eq = fraction(factor(eq))[0] factors = Mul.make_args(factor(eq)) roots = [fac.as_base_exp() for fac in factors if len(fac.args)!=0] if len(roots)>1 or roots[0][1]>1: for base, expo in roots: if base.has(f(x)): self.eqs.append(base) if len(self.eqs)>0: return True roots = solve(eq, df) if len(roots)>0: self.eqs = [(df - root) for root in roots] # Avoid infinite recursion matches = self.eqs != [eq_orig] return matches for i in factors: if i.has(f(x)): self.eqs.append(i) return len(self.eqs)>0 and len(factors)>1 def _get_general_solution(self, *, simplify_flag: bool = True): func = self.ode_problem.func.func x = self.ode_problem.sym eqns = self.eqs sols = [] for eq in eqns: try: sol = dsolve(eq, func(x)) except NotImplementedError: continue else: if isinstance(sol, list): sols.extend(sol) else: sols.append(sol) if sols == []: raise NotImplementedError("The given ODE " + str(eq) + " cannot be solved by" + " the factorable group method") return sols class RiccatiSpecial(SinglePatternODESolver): r""" The general Riccati equation has the form .. math:: dy/dx = f(x) y^2 + g(x) y + h(x)\text{.} While it does not have a general solution [1], the "special" form, `dy/dx = a y^2 - b x^c`, does have solutions in many cases [2]. This routine returns a solution for `a(dy/dx) = b y^2 + c y/x + d/x^2` that is obtained by using a suitable change of variables to reduce it to the special form and is valid when neither `a` nor `b` are zero and either `c` or `d` is zero. >>> from sympy.abc import x, a, b, c, d >>> from sympy import dsolve, checkodesol, pprint, Function >>> f = Function('f') >>> y = f(x) >>> genform = a*y.diff(x) - (b*y**2 + c*y/x + d/x**2) >>> sol = dsolve(genform, y, hint="Riccati_special_minus2") >>> pprint(sol, wrap_line=False) / / __________________ \\ | __________________ | / 2 || | / 2 | \/ 4*b*d - (a + c) *log(x)|| -|a + c - \/ 4*b*d - (a + c) *tan|C1 + ----------------------------|| \ \ 2*a // f(x) = ------------------------------------------------------------------------ 2*b*x >>> checkodesol(genform, sol, order=1)[0] True References ========== - https://www.maplesoft.com/support/help/Maple/view.aspx?path=odeadvisor/Riccati - https://eqworld.ipmnet.ru/en/solutions/ode/ode0106.pdf - https://eqworld.ipmnet.ru/en/solutions/ode/ode0123.pdf """ hint = "Riccati_special_minus2" has_integral = False order = [1] def _wilds(self, f, x, order): a = Wild('a', exclude=[x, f(x), f(x).diff(x), 0]) b = Wild('b', exclude=[x, f(x), f(x).diff(x), 0]) c = Wild('c', exclude=[x, f(x), f(x).diff(x)]) d = Wild('d', exclude=[x, f(x), f(x).diff(x)]) return a, b, c, d def _equation(self, fx, x, order): a, b, c, d = self.wilds() return a*fx.diff(x) + b*fx**2 + c*fx/x + d/x**2 def _get_general_solution(self, *, simplify_flag: bool = True): a, b, c, d = self.wilds_match() fx = self.ode_problem.func x = self.ode_problem.sym (C1,) = self.ode_problem.get_numbered_constants(num=1) mu = sqrt(4*d*b - (a - c)**2) gensol = Eq(fx, (a - c - mu*tan(mu/(2*a)*log(x) + C1))/(2*b*x)) return [gensol] class RationalRiccati(SinglePatternODESolver): r""" Gives general solutions to the first order Riccati differential equations that have atleast one rational particular solution. .. math :: y' = b_0(x) + b_1(x) y + b_2(x) y^2 where `b_0`, `b_1` and `b_2` are rational functions of `x` with `b_2 \ne 0` (`b_2 = 0` would make it a Bernoulli equation). Examples ======== >>> from sympy import Symbol, Function, dsolve, checkodesol >>> f = Function('f') >>> x = Symbol('x') >>> eq = -x**4*f(x)**2 + x**3*f(x).diff(x) + x**2*f(x) + 20 >>> sol = dsolve(eq, hint="1st_rational_riccati") >>> sol Eq(f(x), (4*C1 - 5*x**9 - 4)/(x**2*(C1 + x**9 - 1))) >>> checkodesol(eq, sol) (True, 0) References ========== - Riccati ODE: https://en.wikipedia.org/wiki/Riccati_equation - N. Thieu Vo - Rational and Algebraic Solutions of First-Order Algebraic ODEs: Algorithm 11, pp. 78 - https://www3.risc.jku.at/publications/download/risc_5387/PhDThesisThieu.pdf """ has_integral = False hint = "1st_rational_riccati" order = [1] def _wilds(self, f, x, order): b0 = Wild('b0', exclude=[f(x), f(x).diff(x)]) b1 = Wild('b1', exclude=[f(x), f(x).diff(x)]) b2 = Wild('b2', exclude=[f(x), f(x).diff(x)]) return (b0, b1, b2) def _equation(self, fx, x, order): b0, b1, b2 = self.wilds() return fx.diff(x) - b0 - b1*fx - b2*fx**2 def _matches(self): eq = self.ode_problem.eq_expanded f = self.ode_problem.func.func x = self.ode_problem.sym order = self.ode_problem.order if order != 1: return False match, funcs = match_riccati(eq, f, x) if not match: return False _b0, _b1, _b2 = funcs b0, b1, b2 = self.wilds() self._wilds_match = match = {b0: _b0, b1: _b1, b2: _b2} return True def _get_general_solution(self, *, simplify_flag: bool = True): # Match the equation b0, b1, b2 = self.wilds_match() fx = self.ode_problem.func x = self.ode_problem.sym return solve_riccati(fx, x, b0, b1, b2, gensol=True) class SecondNonlinearAutonomousConserved(SinglePatternODESolver): r""" Gives solution for the autonomous second order nonlinear differential equation of the form .. math :: f''(x) = g(f(x)) The solution for this differential equation can be computed by multiplying by `f'(x)` and integrating on both sides, converting it into a first order differential equation. Examples ======== >>> from sympy import Function, symbols, dsolve >>> f, g = symbols('f g', cls=Function) >>> x = symbols('x') >>> eq = f(x).diff(x, 2) - g(f(x)) >>> dsolve(eq, simplify=False) [Eq(Integral(1/sqrt(C1 + 2*Integral(g(_u), _u)), (_u, f(x))), C2 + x), Eq(Integral(1/sqrt(C1 + 2*Integral(g(_u), _u)), (_u, f(x))), C2 - x)] >>> from sympy import exp, log >>> eq = f(x).diff(x, 2) - exp(f(x)) + log(f(x)) >>> dsolve(eq, simplify=False) [Eq(Integral(1/sqrt(-2*_u*log(_u) + 2*_u + C1 + 2*exp(_u)), (_u, f(x))), C2 + x), Eq(Integral(1/sqrt(-2*_u*log(_u) + 2*_u + C1 + 2*exp(_u)), (_u, f(x))), C2 - x)] References ========== - https://eqworld.ipmnet.ru/en/solutions/ode/ode0301.pdf """ hint = "2nd_nonlinear_autonomous_conserved" has_integral = True order = [2] def _wilds(self, f, x, order): fy = Wild('fy', exclude=[0, f(x).diff(x), f(x).diff(x, 2)]) return (fy, ) def _equation(self, fx, x, order): fy = self.wilds()[0] return fx.diff(x, 2) + fy def _verify(self, fx): return self.ode_problem.is_autonomous def _get_general_solution(self, *, simplify_flag: bool = True): g = self.wilds_match()[0] fx = self.ode_problem.func x = self.ode_problem.sym u = Dummy('u') g = g.subs(fx, u) C1, C2 = self.ode_problem.get_numbered_constants(num=2) inside = -2*Integral(g, u) + C1 lhs = Integral(1/sqrt(inside), (u, fx)) return [Eq(lhs, C2 + x), Eq(lhs, C2 - x)] class Liouville(SinglePatternODESolver): r""" Solves 2nd order Liouville differential equations. The general form of a Liouville ODE is .. math:: \frac{d^2 y}{dx^2} + g(y) \left(\! \frac{dy}{dx}\!\right)^2 + h(x) \frac{dy}{dx}\text{.} The general solution is: >>> from sympy import Function, dsolve, Eq, pprint, diff >>> from sympy.abc import x >>> f, g, h = map(Function, ['f', 'g', 'h']) >>> genform = Eq(diff(f(x),x,x) + g(f(x))*diff(f(x),x)**2 + ... h(x)*diff(f(x),x), 0) >>> pprint(genform) 2 2 /d \ d d g(f(x))*|--(f(x))| + h(x)*--(f(x)) + ---(f(x)) = 0 \dx / dx 2 dx >>> pprint(dsolve(genform, f(x), hint='Liouville_Integral')) f(x) / / | | | / | / | | | | | - | h(x) dx | | g(y) dy | | | | | / | / C1 + C2* | e dx + | e dy = 0 | | / / Examples ======== >>> from sympy import Function, dsolve, Eq, pprint >>> from sympy.abc import x >>> f = Function('f') >>> pprint(dsolve(diff(f(x), x, x) + diff(f(x), x)**2/f(x) + ... diff(f(x), x)/x, f(x), hint='Liouville')) ________________ ________________ [f(x) = -\/ C1 + C2*log(x) , f(x) = \/ C1 + C2*log(x) ] References ========== - Goldstein and Braun, "Advanced Methods for the Solution of Differential Equations", pp. 98 - https://www.maplesoft.com/support/help/Maple/view.aspx?path=odeadvisor/Liouville # indirect doctest """ hint = "Liouville" has_integral = True order = [2] def _wilds(self, f, x, order): d = Wild('d', exclude=[f(x).diff(x), f(x).diff(x, 2)]) e = Wild('e', exclude=[f(x).diff(x)]) k = Wild('k', exclude=[f(x).diff(x)]) return d, e, k def _equation(self, fx, x, order): # Liouville ODE in the form # f(x).diff(x, 2) + g(f(x))*(f(x).diff(x))**2 + h(x)*f(x).diff(x) # See Goldstein and Braun, "Advanced Methods for the Solution of # Differential Equations", pg. 98 d, e, k = self.wilds() return d*fx.diff(x, 2) + e*fx.diff(x)**2 + k*fx.diff(x) def _verify(self, fx): d, e, k = self.wilds_match() self.y = Dummy('y') x = self.ode_problem.sym self.g = simplify(e/d).subs(fx, self.y) self.h = simplify(k/d).subs(fx, self.y) if self.y in self.h.free_symbols or x in self.g.free_symbols: return False return True def _get_general_solution(self, *, simplify_flag: bool = True): d, e, k = self.wilds_match() fx = self.ode_problem.func x = self.ode_problem.sym C1, C2 = self.ode_problem.get_numbered_constants(num=2) int = Integral(exp(Integral(self.g, self.y)), (self.y, None, fx)) gen_sol = Eq(int + C1*Integral(exp(-Integral(self.h, x)), x) + C2, 0) return [gen_sol] class Separable(SinglePatternODESolver): r""" Solves separable 1st order differential equations. This is any differential equation that can be written as `P(y) \tfrac{dy}{dx} = Q(x)`. The solution can then just be found by rearranging terms and integrating: `\int P(y) \,dy = \int Q(x) \,dx`. This hint uses :py:meth:`sympy.simplify.simplify.separatevars` as its back end, so if a separable equation is not caught by this solver, it is most likely the fault of that function. :py:meth:`~sympy.simplify.simplify.separatevars` is smart enough to do most expansion and factoring necessary to convert a separable equation `F(x, y)` into the proper form `P(x)\cdot{}Q(y)`. The general solution is:: >>> from sympy import Function, dsolve, Eq, pprint >>> from sympy.abc import x >>> a, b, c, d, f = map(Function, ['a', 'b', 'c', 'd', 'f']) >>> genform = Eq(a(x)*b(f(x))*f(x).diff(x), c(x)*d(f(x))) >>> pprint(genform) d a(x)*b(f(x))*--(f(x)) = c(x)*d(f(x)) dx >>> pprint(dsolve(genform, f(x), hint='separable_Integral')) f(x) / / | | | b(y) | c(x) | ---- dy = C1 + | ---- dx | d(y) | a(x) | | / / Examples ======== >>> from sympy import Function, dsolve, Eq >>> from sympy.abc import x >>> f = Function('f') >>> pprint(dsolve(Eq(f(x)*f(x).diff(x) + x, 3*x*f(x)**2), f(x), ... hint='separable', simplify=False)) / 2 \ 2 log\3*f (x) - 1/ x ---------------- = C1 + -- 6 2 References ========== - M. Tenenbaum & H. Pollard, "Ordinary Differential Equations", Dover 1963, pp. 52 # indirect doctest """ hint = "separable" has_integral = True order = [1] def _wilds(self, f, x, order): d = Wild('d', exclude=[f(x).diff(x), f(x).diff(x, 2)]) e = Wild('e', exclude=[f(x).diff(x)]) return d, e def _equation(self, fx, x, order): d, e = self.wilds() return d + e*fx.diff(x) def _verify(self, fx): d, e = self.wilds_match() self.y = Dummy('y') x = self.ode_problem.sym d = separatevars(d.subs(fx, self.y)) e = separatevars(e.subs(fx, self.y)) # m1[coeff]*m1[x]*m1[y] + m2[coeff]*m2[x]*m2[y]*y' self.m1 = separatevars(d, dict=True, symbols=(x, self.y)) self.m2 = separatevars(e, dict=True, symbols=(x, self.y)) if self.m1 and self.m2: return True return False def _get_match_object(self): fx = self.ode_problem.func x = self.ode_problem.sym return self.m1, self.m2, x, fx def _get_general_solution(self, *, simplify_flag: bool = True): m1, m2, x, fx = self._get_match_object() (C1,) = self.ode_problem.get_numbered_constants(num=1) int = Integral(m2['coeff']*m2[self.y]/m1[self.y], (self.y, None, fx)) gen_sol = Eq(int, Integral(-m1['coeff']*m1[x]/ m2[x], x) + C1) return [gen_sol] class SeparableReduced(Separable): r""" Solves a differential equation that can be reduced to the separable form. The general form of this equation is .. math:: y' + (y/x) H(x^n y) = 0\text{}. This can be solved by substituting `u(y) = x^n y`. The equation then reduces to the separable form `\frac{u'}{u (\mathrm{power} - H(u))} - \frac{1}{x} = 0`. The general solution is: >>> from sympy import Function, dsolve, pprint >>> from sympy.abc import x, n >>> f, g = map(Function, ['f', 'g']) >>> genform = f(x).diff(x) + (f(x)/x)*g(x**n*f(x)) >>> pprint(genform) / n \ d f(x)*g\x *f(x)/ --(f(x)) + --------------- dx x >>> pprint(dsolve(genform, hint='separable_reduced')) n x *f(x) / | | 1 | ------------ dy = C1 + log(x) | y*(n - g(y)) | / See Also ======== :obj:`sympy.solvers.ode.single.Separable` Examples ======== >>> from sympy import dsolve, Function, pprint >>> from sympy.abc import x >>> f = Function('f') >>> d = f(x).diff(x) >>> eq = (x - x**2*f(x))*d - f(x) >>> dsolve(eq, hint='separable_reduced') [Eq(f(x), (1 - sqrt(C1*x**2 + 1))/x), Eq(f(x), (sqrt(C1*x**2 + 1) + 1)/x)] >>> pprint(dsolve(eq, hint='separable_reduced')) ___________ ___________ / 2 / 2 1 - \/ C1*x + 1 \/ C1*x + 1 + 1 [f(x) = ------------------, f(x) = ------------------] x x References ========== - Joel Moses, "Symbolic Integration - The Stormy Decade", Communications of the ACM, Volume 14, Number 8, August 1971, pp. 558 """ hint = "separable_reduced" has_integral = True order = [1] def _degree(self, expr, x): # Made this function to calculate the degree of # x in an expression. If expr will be of form # x**p*y, (wheare p can be variables/rationals) then it # will return p. for val in expr: if val.has(x): if isinstance(val, Pow) and val.as_base_exp()[0] == x: return (val.as_base_exp()[1]) elif val == x: return (val.as_base_exp()[1]) else: return self._degree(val.args, x) return 0 def _powers(self, expr): # this function will return all the different relative power of x w.r.t f(x). # expr = x**p * f(x)**q then it will return {p/q}. pows = set() fx = self.ode_problem.func x = self.ode_problem.sym self.y = Dummy('y') if isinstance(expr, Add): exprs = expr.atoms(Add) elif isinstance(expr, Mul): exprs = expr.atoms(Mul) elif isinstance(expr, Pow): exprs = expr.atoms(Pow) else: exprs = {expr} for arg in exprs: if arg.has(x): _, u = arg.as_independent(x, fx) pow = self._degree((u.subs(fx, self.y), ), x)/self._degree((u.subs(fx, self.y), ), self.y) pows.add(pow) return pows def _verify(self, fx): num, den = self.wilds_match() x = self.ode_problem.sym factor = simplify(x/fx*num/den) # Try representing factor in terms of x^n*y # where n is lowest power of x in factor; # first remove terms like sqrt(2)*3 from factor.atoms(Mul) num, dem = factor.as_numer_denom() num = expand(num) dem = expand(dem) pows = self._powers(num) pows.update(self._powers(dem)) pows = list(pows) if(len(pows)==1) and pows[0]!=zoo: self.t = Dummy('t') self.r2 = {'t': self.t} num = num.subs(x**pows[0]*fx, self.t) dem = dem.subs(x**pows[0]*fx, self.t) test = num/dem free = test.free_symbols if len(free) == 1 and free.pop() == self.t: self.r2.update({'power' : pows[0], 'u' : test}) return True return False return False def _get_match_object(self): fx = self.ode_problem.func x = self.ode_problem.sym u = self.r2['u'].subs(self.r2['t'], self.y) ycoeff = 1/(self.y*(self.r2['power'] - u)) m1 = {self.y: 1, x: -1/x, 'coeff': 1} m2 = {self.y: ycoeff, x: 1, 'coeff': 1} return m1, m2, x, x**self.r2['power']*fx class HomogeneousCoeffSubsDepDivIndep(SinglePatternODESolver): r""" Solves a 1st order differential equation with homogeneous coefficients using the substitution `u_1 = \frac{\text{}}{\text{}}`. This is a differential equation .. math:: P(x, y) + Q(x, y) dy/dx = 0 such that `P` and `Q` are homogeneous and of the same order. A function `F(x, y)` is homogeneous of order `n` if `F(x t, y t) = t^n F(x, y)`. Equivalently, `F(x, y)` can be rewritten as `G(y/x)` or `H(x/y)`. See also the docstring of :py:meth:`~sympy.solvers.ode.homogeneous_order`. If the coefficients `P` and `Q` in the differential equation above are homogeneous functions of the same order, then it can be shown that the substitution `y = u_1 x` (i.e. `u_1 = y/x`) will turn the differential equation into an equation separable in the variables `x` and `u`. If `h(u_1)` is the function that results from making the substitution `u_1 = f(x)/x` on `P(x, f(x))` and `g(u_2)` is the function that results from the substitution on `Q(x, f(x))` in the differential equation `P(x, f(x)) + Q(x, f(x)) f'(x) = 0`, then the general solution is:: >>> from sympy import Function, dsolve, pprint >>> from sympy.abc import x >>> f, g, h = map(Function, ['f', 'g', 'h']) >>> genform = g(f(x)/x) + h(f(x)/x)*f(x).diff(x) >>> pprint(genform) /f(x)\ /f(x)\ d g|----| + h|----|*--(f(x)) \ x / \ x / dx >>> pprint(dsolve(genform, f(x), ... hint='1st_homogeneous_coeff_subs_dep_div_indep_Integral')) f(x) ---- x / | | -h(u1) log(x) = C1 + | ---------------- d(u1) | u1*h(u1) + g(u1) | / Where `u_1 h(u_1) + g(u_1) \ne 0` and `x \ne 0`. See also the docstrings of :obj:`~sympy.solvers.ode.single.HomogeneousCoeffBest` and :obj:`~sympy.solvers.ode.single.HomogeneousCoeffSubsIndepDivDep`. Examples ======== >>> from sympy import Function, dsolve >>> from sympy.abc import x >>> f = Function('f') >>> pprint(dsolve(2*x*f(x) + (x**2 + f(x)**2)*f(x).diff(x), f(x), ... hint='1st_homogeneous_coeff_subs_dep_div_indep', simplify=False)) / 3 \ |3*f(x) f (x)| log|------ + -----| | x 3 | \ x / log(x) = log(C1) - ------------------- 3 References ========== - https://en.wikipedia.org/wiki/Homogeneous_differential_equation - M. Tenenbaum & H. Pollard, "Ordinary Differential Equations", Dover 1963, pp. 59 # indirect doctest """ hint = "1st_homogeneous_coeff_subs_dep_div_indep" has_integral = True order = [1] def _wilds(self, f, x, order): d = Wild('d', exclude=[f(x).diff(x), f(x).diff(x, 2)]) e = Wild('e', exclude=[f(x).diff(x)]) return d, e def _equation(self, fx, x, order): d, e = self.wilds() return d + e*fx.diff(x) def _verify(self, fx): self.d, self.e = self.wilds_match() self.y = Dummy('y') x = self.ode_problem.sym self.d = separatevars(self.d.subs(fx, self.y)) self.e = separatevars(self.e.subs(fx, self.y)) ordera = homogeneous_order(self.d, x, self.y) orderb = homogeneous_order(self.e, x, self.y) if ordera == orderb and ordera is not None: self.u = Dummy('u') if simplify((self.d + self.u*self.e).subs({x: 1, self.y: self.u})) != 0: return True return False return False def _get_match_object(self): fx = self.ode_problem.func x = self.ode_problem.sym self.u1 = Dummy('u1') xarg = 0 yarg = 0 return [self.d, self.e, fx, x, self.u, self.u1, self.y, xarg, yarg] def _get_general_solution(self, *, simplify_flag: bool = True): d, e, fx, x, u, u1, y, xarg, yarg = self._get_match_object() (C1,) = self.ode_problem.get_numbered_constants(num=1) int = Integral( (-e/(d + u1*e)).subs({x: 1, y: u1}), (u1, None, fx/x)) sol = logcombine(Eq(log(x), int + log(C1)), force=True) gen_sol = sol.subs(fx, u).subs(((u, u - yarg), (x, x - xarg), (u, fx))) return [gen_sol] class HomogeneousCoeffSubsIndepDivDep(SinglePatternODESolver): r""" Solves a 1st order differential equation with homogeneous coefficients using the substitution `u_2 = \frac{\text{}}{\text{}}`. This is a differential equation .. math:: P(x, y) + Q(x, y) dy/dx = 0 such that `P` and `Q` are homogeneous and of the same order. A function `F(x, y)` is homogeneous of order `n` if `F(x t, y t) = t^n F(x, y)`. Equivalently, `F(x, y)` can be rewritten as `G(y/x)` or `H(x/y)`. See also the docstring of :py:meth:`~sympy.solvers.ode.homogeneous_order`. If the coefficients `P` and `Q` in the differential equation above are homogeneous functions of the same order, then it can be shown that the substitution `x = u_2 y` (i.e. `u_2 = x/y`) will turn the differential equation into an equation separable in the variables `y` and `u_2`. If `h(u_2)` is the function that results from making the substitution `u_2 = x/f(x)` on `P(x, f(x))` and `g(u_2)` is the function that results from the substitution on `Q(x, f(x))` in the differential equation `P(x, f(x)) + Q(x, f(x)) f'(x) = 0`, then the general solution is: >>> from sympy import Function, dsolve, pprint >>> from sympy.abc import x >>> f, g, h = map(Function, ['f', 'g', 'h']) >>> genform = g(x/f(x)) + h(x/f(x))*f(x).diff(x) >>> pprint(genform) / x \ / x \ d g|----| + h|----|*--(f(x)) \f(x)/ \f(x)/ dx >>> pprint(dsolve(genform, f(x), ... hint='1st_homogeneous_coeff_subs_indep_div_dep_Integral')) x ---- f(x) / | | -g(u1) | ---------------- d(u1) | u1*g(u1) + h(u1) | / f(x) = C1*e Where `u_1 g(u_1) + h(u_1) \ne 0` and `f(x) \ne 0`. See also the docstrings of :obj:`~sympy.solvers.ode.single.HomogeneousCoeffBest` and :obj:`~sympy.solvers.ode.single.HomogeneousCoeffSubsDepDivIndep`. Examples ======== >>> from sympy import Function, pprint, dsolve >>> from sympy.abc import x >>> f = Function('f') >>> pprint(dsolve(2*x*f(x) + (x**2 + f(x)**2)*f(x).diff(x), f(x), ... hint='1st_homogeneous_coeff_subs_indep_div_dep', ... simplify=False)) / 2 \ | 3*x | log|----- + 1| | 2 | \f (x) / log(f(x)) = log(C1) - -------------- 3 References ========== - https://en.wikipedia.org/wiki/Homogeneous_differential_equation - M. Tenenbaum & H. Pollard, "Ordinary Differential Equations", Dover 1963, pp. 59 # indirect doctest """ hint = "1st_homogeneous_coeff_subs_indep_div_dep" has_integral = True order = [1] def _wilds(self, f, x, order): d = Wild('d', exclude=[f(x).diff(x), f(x).diff(x, 2)]) e = Wild('e', exclude=[f(x).diff(x)]) return d, e def _equation(self, fx, x, order): d, e = self.wilds() return d + e*fx.diff(x) def _verify(self, fx): self.d, self.e = self.wilds_match() self.y = Dummy('y') x = self.ode_problem.sym self.d = separatevars(self.d.subs(fx, self.y)) self.e = separatevars(self.e.subs(fx, self.y)) ordera = homogeneous_order(self.d, x, self.y) orderb = homogeneous_order(self.e, x, self.y) if ordera == orderb and ordera is not None: self.u = Dummy('u') if simplify((self.e + self.u*self.d).subs({x: self.u, self.y: 1})) != 0: return True return False return False def _get_match_object(self): fx = self.ode_problem.func x = self.ode_problem.sym self.u1 = Dummy('u1') xarg = 0 yarg = 0 return [self.d, self.e, fx, x, self.u, self.u1, self.y, xarg, yarg] def _get_general_solution(self, *, simplify_flag: bool = True): d, e, fx, x, u, u1, y, xarg, yarg = self._get_match_object() (C1,) = self.ode_problem.get_numbered_constants(num=1) int = Integral(simplify((-d/(e + u1*d)).subs({x: u1, y: 1})), (u1, None, x/fx)) # type: ignore sol = logcombine(Eq(log(fx), int + log(C1)), force=True) gen_sol = sol.subs(fx, u).subs(((u, u - yarg), (x, x - xarg), (u, fx))) return [gen_sol] class HomogeneousCoeffBest(HomogeneousCoeffSubsIndepDivDep, HomogeneousCoeffSubsDepDivIndep): r""" Returns the best solution to an ODE from the two hints ``1st_homogeneous_coeff_subs_dep_div_indep`` and ``1st_homogeneous_coeff_subs_indep_div_dep``. This is as determined by :py:meth:`~sympy.solvers.ode.ode.ode_sol_simplicity`. See the :obj:`~sympy.solvers.ode.single.HomogeneousCoeffSubsIndepDivDep` and :obj:`~sympy.solvers.ode.single.HomogeneousCoeffSubsDepDivIndep` docstrings for more information on these hints. Note that there is no ``ode_1st_homogeneous_coeff_best_Integral`` hint. Examples ======== >>> from sympy import Function, dsolve, pprint >>> from sympy.abc import x >>> f = Function('f') >>> pprint(dsolve(2*x*f(x) + (x**2 + f(x)**2)*f(x).diff(x), f(x), ... hint='1st_homogeneous_coeff_best', simplify=False)) / 2 \ | 3*x | log|----- + 1| | 2 | \f (x) / log(f(x)) = log(C1) - -------------- 3 References ========== - https://en.wikipedia.org/wiki/Homogeneous_differential_equation - M. Tenenbaum & H. Pollard, "Ordinary Differential Equations", Dover 1963, pp. 59 # indirect doctest """ hint = "1st_homogeneous_coeff_best" has_integral = False order = [1] def _verify(self, fx): if HomogeneousCoeffSubsIndepDivDep._verify(self, fx) and HomogeneousCoeffSubsDepDivIndep._verify(self, fx): return True return False def _get_general_solution(self, *, simplify_flag: bool = True): # There are two substitutions that solve the equation, u1=y/x and u2=x/y # # They produce different integrals, so try them both and see which # # one is easier sol1 = HomogeneousCoeffSubsIndepDivDep._get_general_solution(self) sol2 = HomogeneousCoeffSubsDepDivIndep._get_general_solution(self) fx = self.ode_problem.func if simplify_flag: sol1 = odesimp(self.ode_problem.eq, *sol1, fx, "1st_homogeneous_coeff_subs_indep_div_dep") sol2 = odesimp(self.ode_problem.eq, *sol2, fx, "1st_homogeneous_coeff_subs_dep_div_indep") return min([sol1, sol2], key=lambda x: ode_sol_simplicity(x, fx, trysolving=not simplify)) class LinearCoefficients(HomogeneousCoeffBest): r""" Solves a differential equation with linear coefficients. The general form of a differential equation with linear coefficients is .. math:: y' + F\left(\!\frac{a_1 x + b_1 y + c_1}{a_2 x + b_2 y + c_2}\!\right) = 0\text{,} where `a_1`, `b_1`, `c_1`, `a_2`, `b_2`, `c_2` are constants and `a_1 b_2 - a_2 b_1 \ne 0`. This can be solved by substituting: .. math:: x = x' + \frac{b_2 c_1 - b_1 c_2}{a_2 b_1 - a_1 b_2} y = y' + \frac{a_1 c_2 - a_2 c_1}{a_2 b_1 - a_1 b_2}\text{.} This substitution reduces the equation to a homogeneous differential equation. See Also ======== :obj:`sympy.solvers.ode.single.HomogeneousCoeffBest` :obj:`sympy.solvers.ode.single.HomogeneousCoeffSubsIndepDivDep` :obj:`sympy.solvers.ode.single.HomogeneousCoeffSubsDepDivIndep` Examples ======== >>> from sympy import dsolve, Function, pprint >>> from sympy.abc import x >>> f = Function('f') >>> df = f(x).diff(x) >>> eq = (x + f(x) + 1)*df + (f(x) - 6*x + 1) >>> dsolve(eq, hint='linear_coefficients') [Eq(f(x), -x - sqrt(C1 + 7*x**2) - 1), Eq(f(x), -x + sqrt(C1 + 7*x**2) - 1)] >>> pprint(dsolve(eq, hint='linear_coefficients')) ___________ ___________ / 2 / 2 [f(x) = -x - \/ C1 + 7*x - 1, f(x) = -x + \/ C1 + 7*x - 1] References ========== - Joel Moses, "Symbolic Integration - The Stormy Decade", Communications of the ACM, Volume 14, Number 8, August 1971, pp. 558 """ hint = "linear_coefficients" has_integral = True order = [1] def _wilds(self, f, x, order): d = Wild('d', exclude=[f(x).diff(x), f(x).diff(x, 2)]) e = Wild('e', exclude=[f(x).diff(x)]) return d, e def _equation(self, fx, x, order): d, e = self.wilds() return d + e*fx.diff(x) def _verify(self, fx): self.d, self.e = self.wilds_match() a, b = self.wilds() F = self.d/self.e x = self.ode_problem.sym params = self._linear_coeff_match(F, fx) if params: self.xarg, self.yarg = params u = Dummy('u') t = Dummy('t') self.y = Dummy('y') # Dummy substitution for df and f(x). dummy_eq = self.ode_problem.eq.subs(((fx.diff(x), t), (fx, u))) reps = ((x, x + self.xarg), (u, u + self.yarg), (t, fx.diff(x)), (u, fx)) dummy_eq = simplify(dummy_eq.subs(reps)) # get the re-cast values for e and d r2 = collect(expand(dummy_eq), [fx.diff(x), fx]).match(a*fx.diff(x) + b) if r2: self.d, self.e = r2[b], r2[a] orderd = homogeneous_order(self.d, x, fx) ordere = homogeneous_order(self.e, x, fx) if orderd == ordere and orderd is not None: self.d = self.d.subs(fx, self.y) self.e = self.e.subs(fx, self.y) return True return False return False def _linear_coeff_match(self, expr, func): r""" Helper function to match hint ``linear_coefficients``. Matches the expression to the form `(a_1 x + b_1 f(x) + c_1)/(a_2 x + b_2 f(x) + c_2)` where the following conditions hold: 1. `a_1`, `b_1`, `c_1`, `a_2`, `b_2`, `c_2` are Rationals; 2. `c_1` or `c_2` are not equal to zero; 3. `a_2 b_1 - a_1 b_2` is not equal to zero. Return ``xarg``, ``yarg`` where 1. ``xarg`` = `(b_2 c_1 - b_1 c_2)/(a_2 b_1 - a_1 b_2)` 2. ``yarg`` = `(a_1 c_2 - a_2 c_1)/(a_2 b_1 - a_1 b_2)` Examples ======== >>> from sympy import Function, sin >>> from sympy.abc import x >>> from sympy.solvers.ode.single import LinearCoefficients >>> f = Function('f') >>> eq = (-25*f(x) - 8*x + 62)/(4*f(x) + 11*x - 11) >>> obj = LinearCoefficients(eq) >>> obj._linear_coeff_match(eq, f(x)) (1/9, 22/9) >>> eq = sin((-5*f(x) - 8*x + 6)/(4*f(x) + x - 1)) >>> obj = LinearCoefficients(eq) >>> obj._linear_coeff_match(eq, f(x)) (19/27, 2/27) >>> eq = sin(f(x)/x) >>> obj = LinearCoefficients(eq) >>> obj._linear_coeff_match(eq, f(x)) """ f = func.func x = func.args[0] def abc(eq): r''' Internal function of _linear_coeff_match that returns Rationals a, b, c if eq is a*x + b*f(x) + c, else None. ''' eq = _mexpand(eq) c = eq.as_independent(x, f(x), as_Add=True)[0] if not c.is_Rational: return a = eq.coeff(x) if not a.is_Rational: return b = eq.coeff(f(x)) if not b.is_Rational: return if eq == a*x + b*f(x) + c: return a, b, c def match(arg): r''' Internal function of _linear_coeff_match that returns Rationals a1, b1, c1, a2, b2, c2 and a2*b1 - a1*b2 of the expression (a1*x + b1*f(x) + c1)/(a2*x + b2*f(x) + c2) if one of c1 or c2 and a2*b1 - a1*b2 is non-zero, else None. ''' n, d = arg.together().as_numer_denom() m = abc(n) if m is not None: a1, b1, c1 = m m = abc(d) if m is not None: a2, b2, c2 = m d = a2*b1 - a1*b2 if (c1 or c2) and d: return a1, b1, c1, a2, b2, c2, d m = [fi.args[0] for fi in expr.atoms(Function) if fi.func != f and len(fi.args) == 1 and not fi.args[0].is_Function] or {expr} m1 = match(m.pop()) if m1 and all(match(mi) == m1 for mi in m): a1, b1, c1, a2, b2, c2, denom = m1 return (b2*c1 - b1*c2)/denom, (a1*c2 - a2*c1)/denom def _get_match_object(self): fx = self.ode_problem.func x = self.ode_problem.sym self.u1 = Dummy('u1') u = Dummy('u') return [self.d, self.e, fx, x, u, self.u1, self.y, self.xarg, self.yarg] class NthOrderReducible(SingleODESolver): r""" Solves ODEs that only involve derivatives of the dependent variable using a substitution of the form `f^n(x) = g(x)`. For example any second order ODE of the form `f''(x) = h(f'(x), x)` can be transformed into a pair of 1st order ODEs `g'(x) = h(g(x), x)` and `f'(x) = g(x)`. Usually the 1st order ODE for `g` is easier to solve. If that gives an explicit solution for `g` then `f` is found simply by integration. Examples ======== >>> from sympy import Function, dsolve, Eq >>> from sympy.abc import x >>> f = Function('f') >>> eq = Eq(x*f(x).diff(x)**2 + f(x).diff(x, 2), 0) >>> dsolve(eq, f(x), hint='nth_order_reducible') ... # doctest: +NORMALIZE_WHITESPACE Eq(f(x), C1 - sqrt(-1/C2)*log(-C2*sqrt(-1/C2) + x) + sqrt(-1/C2)*log(C2*sqrt(-1/C2) + x)) """ hint = "nth_order_reducible" has_integral = False def _matches(self): # Any ODE that can be solved with a substitution and # repeated integration e.g.: # `d^2/dx^2(y) + x*d/dx(y) = constant #f'(x) must be finite for this to work eq = self.ode_problem.eq_preprocessed func = self.ode_problem.func x = self.ode_problem.sym r""" Matches any differential equation that can be rewritten with a smaller order. Only derivatives of ``func`` alone, wrt a single variable, are considered, and only in them should ``func`` appear. """ # ODE only handles functions of 1 variable so this affirms that state assert len(func.args) == 1 vc = [d.variable_count[0] for d in eq.atoms(Derivative) if d.expr == func and len(d.variable_count) == 1] ords = [c for v, c in vc if v == x] if len(ords) < 2: return False self.smallest = min(ords) # make sure func does not appear outside of derivatives D = Dummy() if eq.subs(func.diff(x, self.smallest), D).has(func): return False return True def _get_general_solution(self, *, simplify_flag: bool = True): eq = self.ode_problem.eq f = self.ode_problem.func.func x = self.ode_problem.sym n = self.smallest # get a unique function name for g names = [a.name for a in eq.atoms(AppliedUndef)] while True: name = Dummy().name if name not in names: g = Function(name) break w = f(x).diff(x, n) geq = eq.subs(w, g(x)) gsol = dsolve(geq, g(x)) if not isinstance(gsol, list): gsol = [gsol] # Might be multiple solutions to the reduced ODE: fsol = [] for gsoli in gsol: fsoli = dsolve(gsoli.subs(g(x), w), f(x)) # or do integration n times fsol.append(fsoli) return fsol class SecondHypergeometric(SingleODESolver): r""" Solves 2nd order linear differential equations. It computes special function solutions which can be expressed using the 2F1, 1F1 or 0F1 hypergeometric functions. .. math:: y'' + A(x) y' + B(x) y = 0\text{,} where `A` and `B` are rational functions. These kinds of differential equations have solution of non-Liouvillian form. Given linear ODE can be obtained from 2F1 given by .. math:: (x^2 - x) y'' + ((a + b + 1) x - c) y' + b a y = 0\text{,} where {a, b, c} are arbitrary constants. Notes ===== The algorithm should find any solution of the form .. math:: y = P(x) _pF_q(..; ..;\frac{\alpha x^k + \beta}{\gamma x^k + \delta})\text{,} where pFq is any of 2F1, 1F1 or 0F1 and `P` is an "arbitrary function". Currently only the 2F1 case is implemented in SymPy but the other cases are described in the paper and could be implemented in future (contributions welcome!). Examples ======== >>> from sympy import Function, dsolve, pprint >>> from sympy.abc import x >>> f = Function('f') >>> eq = (x*x - x)*f(x).diff(x,2) + (5*x - 1)*f(x).diff(x) + 4*f(x) >>> pprint(dsolve(eq, f(x), '2nd_hypergeometric')) _ / / 4 \\ |_ /-1, -1 | \ |C1 + C2*|log(x) + -----||* | | | x| \ \ x + 1// 2 1 \ 1 | / f(x) = -------------------------------------------- 3 (x - 1) References ========== - "Non-Liouvillian solutions for second order linear ODEs" by L. Chan, E.S. Cheb-Terrab """ hint = "2nd_hypergeometric" has_integral = True def _matches(self): eq = self.ode_problem.eq_preprocessed func = self.ode_problem.func r = match_2nd_hypergeometric(eq, func) self.match_object = None if r: A, B = r d = equivalence_hypergeometric(A, B, func) if d: if d['type'] == "2F1": self.match_object = match_2nd_2F1_hypergeometric(d['I0'], d['k'], d['sing_point'], func) if self.match_object is not None: self.match_object.update({'A':A, 'B':B}) # We can extend it for 1F1 and 0F1 type also. return self.match_object is not None def _get_general_solution(self, *, simplify_flag: bool = True): eq = self.ode_problem.eq func = self.ode_problem.func if self.match_object['type'] == "2F1": sol = get_sol_2F1_hypergeometric(eq, func, self.match_object) if sol is None: raise NotImplementedError("The given ODE " + str(eq) + " cannot be solved by" + " the hypergeometric method") return [sol] class NthLinearConstantCoeffHomogeneous(SingleODESolver): r""" Solves an `n`\th order linear homogeneous differential equation with constant coefficients. This is an equation of the form .. math:: a_n f^{(n)}(x) + a_{n-1} f^{(n-1)}(x) + \cdots + a_1 f'(x) + a_0 f(x) = 0\text{.} These equations can be solved in a general manner, by taking the roots of the characteristic equation `a_n m^n + a_{n-1} m^{n-1} + \cdots + a_1 m + a_0 = 0`. The solution will then be the sum of `C_n x^i e^{r x}` terms, for each where `C_n` is an arbitrary constant, `r` is a root of the characteristic equation and `i` is one of each from 0 to the multiplicity of the root - 1 (for example, a root 3 of multiplicity 2 would create the terms `C_1 e^{3 x} + C_2 x e^{3 x}`). The exponential is usually expanded for complex roots using Euler's equation `e^{I x} = \cos(x) + I \sin(x)`. Complex roots always come in conjugate pairs in polynomials with real coefficients, so the two roots will be represented (after simplifying the constants) as `e^{a x} \left(C_1 \cos(b x) + C_2 \sin(b x)\right)`. If SymPy cannot find exact roots to the characteristic equation, a :py:class:`~sympy.polys.rootoftools.ComplexRootOf` instance will be return instead. >>> from sympy import Function, dsolve >>> from sympy.abc import x >>> f = Function('f') >>> dsolve(f(x).diff(x, 5) + 10*f(x).diff(x) - 2*f(x), f(x), ... hint='nth_linear_constant_coeff_homogeneous') ... # doctest: +NORMALIZE_WHITESPACE Eq(f(x), C5*exp(x*CRootOf(_x**5 + 10*_x - 2, 0)) + (C1*sin(x*im(CRootOf(_x**5 + 10*_x - 2, 1))) + C2*cos(x*im(CRootOf(_x**5 + 10*_x - 2, 1))))*exp(x*re(CRootOf(_x**5 + 10*_x - 2, 1))) + (C3*sin(x*im(CRootOf(_x**5 + 10*_x - 2, 3))) + C4*cos(x*im(CRootOf(_x**5 + 10*_x - 2, 3))))*exp(x*re(CRootOf(_x**5 + 10*_x - 2, 3)))) Note that because this method does not involve integration, there is no ``nth_linear_constant_coeff_homogeneous_Integral`` hint. Examples ======== >>> from sympy import Function, dsolve, pprint >>> from sympy.abc import x >>> f = Function('f') >>> pprint(dsolve(f(x).diff(x, 4) + 2*f(x).diff(x, 3) - ... 2*f(x).diff(x, 2) - 6*f(x).diff(x) + 5*f(x), f(x), ... hint='nth_linear_constant_coeff_homogeneous')) x -2*x f(x) = (C1 + C2*x)*e + (C3*sin(x) + C4*cos(x))*e References ========== - https://en.wikipedia.org/wiki/Linear_differential_equation section: Nonhomogeneous_equation_with_constant_coefficients - M. Tenenbaum & H. Pollard, "Ordinary Differential Equations", Dover 1963, pp. 211 # indirect doctest """ hint = "nth_linear_constant_coeff_homogeneous" has_integral = False def _matches(self): eq = self.ode_problem.eq_high_order_free func = self.ode_problem.func order = self.ode_problem.order x = self.ode_problem.sym self.r = self.ode_problem.get_linear_coefficients(eq, func, order) if order and self.r and not any(self.r[i].has(x) for i in self.r if i >= 0): if not self.r[-1]: return True else: return False return False def _get_general_solution(self, *, simplify_flag: bool = True): fx = self.ode_problem.func order = self.ode_problem.order roots, collectterms = _get_const_characteristic_eq_sols(self.r, fx, order) # A generator of constants constants = self.ode_problem.get_numbered_constants(num=len(roots)) gsol = Add(*[i*j for (i, j) in zip(constants, roots)]) gsol = Eq(fx, gsol) if simplify_flag: gsol = _get_simplified_sol([gsol], fx, collectterms) return [gsol] class NthLinearConstantCoeffVariationOfParameters(SingleODESolver): r""" Solves an `n`\th order linear differential equation with constant coefficients using the method of variation of parameters. This method works on any differential equations of the form .. math:: f^{(n)}(x) + a_{n-1} f^{(n-1)}(x) + \cdots + a_1 f'(x) + a_0 f(x) = P(x)\text{.} This method works by assuming that the particular solution takes the form .. math:: \sum_{x=1}^{n} c_i(x) y_i(x)\text{,} where `y_i` is the `i`\th solution to the homogeneous equation. The solution is then solved using Wronskian's and Cramer's Rule. The particular solution is given by .. math:: \sum_{x=1}^n \left( \int \frac{W_i(x)}{W(x)} \,dx \right) y_i(x) \text{,} where `W(x)` is the Wronskian of the fundamental system (the system of `n` linearly independent solutions to the homogeneous equation), and `W_i(x)` is the Wronskian of the fundamental system with the `i`\th column replaced with `[0, 0, \cdots, 0, P(x)]`. This method is general enough to solve any `n`\th order inhomogeneous linear differential equation with constant coefficients, but sometimes SymPy cannot simplify the Wronskian well enough to integrate it. If this method hangs, try using the ``nth_linear_constant_coeff_variation_of_parameters_Integral`` hint and simplifying the integrals manually. Also, prefer using ``nth_linear_constant_coeff_undetermined_coefficients`` when it applies, because it does not use integration, making it faster and more reliable. Warning, using simplify=False with 'nth_linear_constant_coeff_variation_of_parameters' in :py:meth:`~sympy.solvers.ode.dsolve` may cause it to hang, because it will not attempt to simplify the Wronskian before integrating. It is recommended that you only use simplify=False with 'nth_linear_constant_coeff_variation_of_parameters_Integral' for this method, especially if the solution to the homogeneous equation has trigonometric functions in it. Examples ======== >>> from sympy import Function, dsolve, pprint, exp, log >>> from sympy.abc import x >>> f = Function('f') >>> pprint(dsolve(f(x).diff(x, 3) - 3*f(x).diff(x, 2) + ... 3*f(x).diff(x) - f(x) - exp(x)*log(x), f(x), ... hint='nth_linear_constant_coeff_variation_of_parameters')) / / / x*log(x) 11*x\\\ x f(x) = |C1 + x*|C2 + x*|C3 + -------- - ----|||*e \ \ \ 6 36 /// References ========== - https://en.wikipedia.org/wiki/Variation_of_parameters - https://planetmath.org/VariationOfParameters - M. Tenenbaum & H. Pollard, "Ordinary Differential Equations", Dover 1963, pp. 233 # indirect doctest """ hint = "nth_linear_constant_coeff_variation_of_parameters" has_integral = True def _matches(self): eq = self.ode_problem.eq_high_order_free func = self.ode_problem.func order = self.ode_problem.order x = self.ode_problem.sym self.r = self.ode_problem.get_linear_coefficients(eq, func, order) if order and self.r and not any(self.r[i].has(x) for i in self.r if i >= 0): if self.r[-1]: return True else: return False return False def _get_general_solution(self, *, simplify_flag: bool = True): eq = self.ode_problem.eq_high_order_free f = self.ode_problem.func.func x = self.ode_problem.sym order = self.ode_problem.order roots, collectterms = _get_const_characteristic_eq_sols(self.r, f(x), order) # A generator of constants constants = self.ode_problem.get_numbered_constants(num=len(roots)) homogen_sol = Add(*[i*j for (i, j) in zip(constants, roots)]) homogen_sol = Eq(f(x), homogen_sol) homogen_sol = _solve_variation_of_parameters(eq, f(x), roots, homogen_sol, order, self.r, simplify_flag) if simplify_flag: homogen_sol = _get_simplified_sol([homogen_sol], f(x), collectterms) return [homogen_sol] class NthLinearConstantCoeffUndeterminedCoefficients(SingleODESolver): r""" Solves an `n`\th order linear differential equation with constant coefficients using the method of undetermined coefficients. This method works on differential equations of the form .. math:: a_n f^{(n)}(x) + a_{n-1} f^{(n-1)}(x) + \cdots + a_1 f'(x) + a_0 f(x) = P(x)\text{,} where `P(x)` is a function that has a finite number of linearly independent derivatives. Functions that fit this requirement are finite sums functions of the form `a x^i e^{b x} \sin(c x + d)` or `a x^i e^{b x} \cos(c x + d)`, where `i` is a non-negative integer and `a`, `b`, `c`, and `d` are constants. For example any polynomial in `x`, functions like `x^2 e^{2 x}`, `x \sin(x)`, and `e^x \cos(x)` can all be used. Products of `\sin`'s and `\cos`'s have a finite number of derivatives, because they can be expanded into `\sin(a x)` and `\cos(b x)` terms. However, SymPy currently cannot do that expansion, so you will need to manually rewrite the expression in terms of the above to use this method. So, for example, you will need to manually convert `\sin^2(x)` into `(1 + \cos(2 x))/2` to properly apply the method of undetermined coefficients on it. This method works by creating a trial function from the expression and all of its linear independent derivatives and substituting them into the original ODE. The coefficients for each term will be a system of linear equations, which are be solved for and substituted, giving the solution. If any of the trial functions are linearly dependent on the solution to the homogeneous equation, they are multiplied by sufficient `x` to make them linearly independent. Examples ======== >>> from sympy import Function, dsolve, pprint, exp, cos >>> from sympy.abc import x >>> f = Function('f') >>> pprint(dsolve(f(x).diff(x, 2) + 2*f(x).diff(x) + f(x) - ... 4*exp(-x)*x**2 + cos(2*x), f(x), ... hint='nth_linear_constant_coeff_undetermined_coefficients')) / / 3\\ | | x || -x 4*sin(2*x) 3*cos(2*x) f(x) = |C1 + x*|C2 + --||*e - ---------- + ---------- \ \ 3 // 25 25 References ========== - https://en.wikipedia.org/wiki/Method_of_undetermined_coefficients - M. Tenenbaum & H. Pollard, "Ordinary Differential Equations", Dover 1963, pp. 221 # indirect doctest """ hint = "nth_linear_constant_coeff_undetermined_coefficients" has_integral = False def _matches(self): eq = self.ode_problem.eq_high_order_free func = self.ode_problem.func order = self.ode_problem.order x = self.ode_problem.sym self.r = self.ode_problem.get_linear_coefficients(eq, func, order) does_match = False if order and self.r and not any(self.r[i].has(x) for i in self.r if i >= 0): if self.r[-1]: eq_homogeneous = Add(eq, -self.r[-1]) undetcoeff = _undetermined_coefficients_match(self.r[-1], x, func, eq_homogeneous) if undetcoeff['test']: self.trialset = undetcoeff['trialset'] does_match = True return does_match def _get_general_solution(self, *, simplify_flag: bool = True): eq = self.ode_problem.eq f = self.ode_problem.func.func x = self.ode_problem.sym order = self.ode_problem.order roots, collectterms = _get_const_characteristic_eq_sols(self.r, f(x), order) # A generator of constants constants = self.ode_problem.get_numbered_constants(num=len(roots)) homogen_sol = Add(*[i*j for (i, j) in zip(constants, roots)]) homogen_sol = Eq(f(x), homogen_sol) self.r.update({'list': roots, 'sol': homogen_sol, 'simpliy_flag': simplify_flag}) gsol = _solve_undetermined_coefficients(eq, f(x), order, self.r, self.trialset) if simplify_flag: gsol = _get_simplified_sol([gsol], f(x), collectterms) return [gsol] class NthLinearEulerEqHomogeneous(SingleODESolver): r""" Solves an `n`\th order linear homogeneous variable-coefficient Cauchy-Euler equidimensional ordinary differential equation. This is an equation with form `0 = a_0 f(x) + a_1 x f'(x) + a_2 x^2 f''(x) \cdots`. These equations can be solved in a general manner, by substituting solutions of the form `f(x) = x^r`, and deriving a characteristic equation for `r`. When there are repeated roots, we include extra terms of the form `C_{r k} \ln^k(x) x^r`, where `C_{r k}` is an arbitrary integration constant, `r` is a root of the characteristic equation, and `k` ranges over the multiplicity of `r`. In the cases where the roots are complex, solutions of the form `C_1 x^a \sin(b \log(x)) + C_2 x^a \cos(b \log(x))` are returned, based on expansions with Euler's formula. The general solution is the sum of the terms found. If SymPy cannot find exact roots to the characteristic equation, a :py:obj:`~.ComplexRootOf` instance will be returned instead. >>> from sympy import Function, dsolve >>> from sympy.abc import x >>> f = Function('f') >>> dsolve(4*x**2*f(x).diff(x, 2) + f(x), f(x), ... hint='nth_linear_euler_eq_homogeneous') ... # doctest: +NORMALIZE_WHITESPACE Eq(f(x), sqrt(x)*(C1 + C2*log(x))) Note that because this method does not involve integration, there is no ``nth_linear_euler_eq_homogeneous_Integral`` hint. The following is for internal use: - ``returns = 'sol'`` returns the solution to the ODE. - ``returns = 'list'`` returns a list of linearly independent solutions, corresponding to the fundamental solution set, for use with non homogeneous solution methods like variation of parameters and undetermined coefficients. Note that, though the solutions should be linearly independent, this function does not explicitly check that. You can do ``assert simplify(wronskian(sollist)) != 0`` to check for linear independence. Also, ``assert len(sollist) == order`` will need to pass. - ``returns = 'both'``, return a dictionary ``{'sol': , 'list': }``. Examples ======== >>> from sympy import Function, dsolve, pprint >>> from sympy.abc import x >>> f = Function('f') >>> eq = f(x).diff(x, 2)*x**2 - 4*f(x).diff(x)*x + 6*f(x) >>> pprint(dsolve(eq, f(x), ... hint='nth_linear_euler_eq_homogeneous')) 2 f(x) = x *(C1 + C2*x) References ========== - https://en.wikipedia.org/wiki/Cauchy%E2%80%93Euler_equation - C. Bender & S. Orszag, "Advanced Mathematical Methods for Scientists and Engineers", Springer 1999, pp. 12 # indirect doctest """ hint = "nth_linear_euler_eq_homogeneous" has_integral = False def _matches(self): eq = self.ode_problem.eq_preprocessed f = self.ode_problem.func.func order = self.ode_problem.order x = self.ode_problem.sym match = self.ode_problem.get_linear_coefficients(eq, f(x), order) self.r = None does_match = False if order and match: coeff = match[order] factor = x**order / coeff self.r = {i: factor*match[i] for i in match} if self.r and all(_test_term(self.r[i], f(x), i) for i in self.r if i >= 0): if not self.r[-1]: does_match = True return does_match def _get_general_solution(self, *, simplify_flag: bool = True): fx = self.ode_problem.func eq = self.ode_problem.eq homogen_sol = _get_euler_characteristic_eq_sols(eq, fx, self.r)[0] return [homogen_sol] class NthLinearEulerEqNonhomogeneousVariationOfParameters(SingleODESolver): r""" Solves an `n`\th order linear non homogeneous Cauchy-Euler equidimensional ordinary differential equation using variation of parameters. This is an equation with form `g(x) = a_0 f(x) + a_1 x f'(x) + a_2 x^2 f''(x) \cdots`. This method works by assuming that the particular solution takes the form .. math:: \sum_{x=1}^{n} c_i(x) y_i(x) {a_n} {x^n} \text{, } where `y_i` is the `i`\th solution to the homogeneous equation. The solution is then solved using Wronskian's and Cramer's Rule. The particular solution is given by multiplying eq given below with `a_n x^{n}` .. math:: \sum_{x=1}^n \left( \int \frac{W_i(x)}{W(x)} \, dx \right) y_i(x) \text{, } where `W(x)` is the Wronskian of the fundamental system (the system of `n` linearly independent solutions to the homogeneous equation), and `W_i(x)` is the Wronskian of the fundamental system with the `i`\th column replaced with `[0, 0, \cdots, 0, \frac{x^{- n}}{a_n} g{\left(x \right)}]`. This method is general enough to solve any `n`\th order inhomogeneous linear differential equation, but sometimes SymPy cannot simplify the Wronskian well enough to integrate it. If this method hangs, try using the ``nth_linear_constant_coeff_variation_of_parameters_Integral`` hint and simplifying the integrals manually. Also, prefer using ``nth_linear_constant_coeff_undetermined_coefficients`` when it applies, because it does not use integration, making it faster and more reliable. Warning, using simplify=False with 'nth_linear_constant_coeff_variation_of_parameters' in :py:meth:`~sympy.solvers.ode.dsolve` may cause it to hang, because it will not attempt to simplify the Wronskian before integrating. It is recommended that you only use simplify=False with 'nth_linear_constant_coeff_variation_of_parameters_Integral' for this method, especially if the solution to the homogeneous equation has trigonometric functions in it. Examples ======== >>> from sympy import Function, dsolve, Derivative >>> from sympy.abc import x >>> f = Function('f') >>> eq = x**2*Derivative(f(x), x, x) - 2*x*Derivative(f(x), x) + 2*f(x) - x**4 >>> dsolve(eq, f(x), ... hint='nth_linear_euler_eq_nonhomogeneous_variation_of_parameters').expand() Eq(f(x), C1*x + C2*x**2 + x**4/6) """ hint = "nth_linear_euler_eq_nonhomogeneous_variation_of_parameters" has_integral = True def _matches(self): eq = self.ode_problem.eq_preprocessed f = self.ode_problem.func.func order = self.ode_problem.order x = self.ode_problem.sym match = self.ode_problem.get_linear_coefficients(eq, f(x), order) self.r = None does_match = False if order and match: coeff = match[order] factor = x**order / coeff self.r = {i: factor*match[i] for i in match} if self.r and all(_test_term(self.r[i], f(x), i) for i in self.r if i >= 0): if self.r[-1]: does_match = True return does_match def _get_general_solution(self, *, simplify_flag: bool = True): eq = self.ode_problem.eq f = self.ode_problem.func.func x = self.ode_problem.sym order = self.ode_problem.order homogen_sol, roots = _get_euler_characteristic_eq_sols(eq, f(x), self.r) self.r[-1] = self.r[-1]/self.r[order] sol = _solve_variation_of_parameters(eq, f(x), roots, homogen_sol, order, self.r, simplify_flag) return [Eq(f(x), homogen_sol.rhs + (sol.rhs - homogen_sol.rhs)*self.r[order])] class NthLinearEulerEqNonhomogeneousUndeterminedCoefficients(SingleODESolver): r""" Solves an `n`\th order linear non homogeneous Cauchy-Euler equidimensional ordinary differential equation using undetermined coefficients. This is an equation with form `g(x) = a_0 f(x) + a_1 x f'(x) + a_2 x^2 f''(x) \cdots`. These equations can be solved in a general manner, by substituting solutions of the form `x = exp(t)`, and deriving a characteristic equation of form `g(exp(t)) = b_0 f(t) + b_1 f'(t) + b_2 f''(t) \cdots` which can be then solved by nth_linear_constant_coeff_undetermined_coefficients if g(exp(t)) has finite number of linearly independent derivatives. Functions that fit this requirement are finite sums functions of the form `a x^i e^{b x} \sin(c x + d)` or `a x^i e^{b x} \cos(c x + d)`, where `i` is a non-negative integer and `a`, `b`, `c`, and `d` are constants. For example any polynomial in `x`, functions like `x^2 e^{2 x}`, `x \sin(x)`, and `e^x \cos(x)` can all be used. Products of `\sin`'s and `\cos`'s have a finite number of derivatives, because they can be expanded into `\sin(a x)` and `\cos(b x)` terms. However, SymPy currently cannot do that expansion, so you will need to manually rewrite the expression in terms of the above to use this method. So, for example, you will need to manually convert `\sin^2(x)` into `(1 + \cos(2 x))/2` to properly apply the method of undetermined coefficients on it. After replacement of x by exp(t), this method works by creating a trial function from the expression and all of its linear independent derivatives and substituting them into the original ODE. The coefficients for each term will be a system of linear equations, which are be solved for and substituted, giving the solution. If any of the trial functions are linearly dependent on the solution to the homogeneous equation, they are multiplied by sufficient `x` to make them linearly independent. Examples ======== >>> from sympy import dsolve, Function, Derivative, log >>> from sympy.abc import x >>> f = Function('f') >>> eq = x**2*Derivative(f(x), x, x) - 2*x*Derivative(f(x), x) + 2*f(x) - log(x) >>> dsolve(eq, f(x), ... hint='nth_linear_euler_eq_nonhomogeneous_undetermined_coefficients').expand() Eq(f(x), C1*x + C2*x**2 + log(x)/2 + 3/4) """ hint = "nth_linear_euler_eq_nonhomogeneous_undetermined_coefficients" has_integral = False def _matches(self): eq = self.ode_problem.eq_high_order_free f = self.ode_problem.func.func order = self.ode_problem.order x = self.ode_problem.sym match = self.ode_problem.get_linear_coefficients(eq, f(x), order) self.r = None does_match = False if order and match: coeff = match[order] factor = x**order / coeff self.r = {i: factor*match[i] for i in match} if self.r and all(_test_term(self.r[i], f(x), i) for i in self.r if i >= 0): if self.r[-1]: e, re = posify(self.r[-1].subs(x, exp(x))) undetcoeff = _undetermined_coefficients_match(e.subs(re), x) if undetcoeff['test']: does_match = True return does_match def _get_general_solution(self, *, simplify_flag: bool = True): f = self.ode_problem.func.func x = self.ode_problem.sym chareq, eq, symbol = S.Zero, S.Zero, Dummy('x') for i in self.r.keys(): if i >= 0: chareq += (self.r[i]*diff(x**symbol, x, i)*x**-symbol).expand() for i in range(1, degree(Poly(chareq, symbol))+1): eq += chareq.coeff(symbol**i)*diff(f(x), x, i) if chareq.as_coeff_add(symbol)[0]: eq += chareq.as_coeff_add(symbol)[0]*f(x) e, re = posify(self.r[-1].subs(x, exp(x))) eq += e.subs(re) self.const_undet_instance = NthLinearConstantCoeffUndeterminedCoefficients(SingleODEProblem(eq, f(x), x)) sol = self.const_undet_instance.get_general_solution(simplify = simplify_flag)[0] sol = sol.subs(x, log(x)) sol = sol.subs(f(log(x)), f(x)).expand() return [sol] class SecondLinearBessel(SingleODESolver): r""" Gives solution of the Bessel differential equation .. math :: x^2 \frac{d^2y}{dx^2} + x \frac{dy}{dx} y(x) + (x^2-n^2) y(x) if `n` is integer then the solution is of the form ``Eq(f(x), C0 besselj(n,x) + C1 bessely(n,x))`` as both the solutions are linearly independent else if `n` is a fraction then the solution is of the form ``Eq(f(x), C0 besselj(n,x) + C1 besselj(-n,x))`` which can also transform into ``Eq(f(x), C0 besselj(n,x) + C1 bessely(n,x))``. Examples ======== >>> from sympy.abc import x >>> from sympy import Symbol >>> v = Symbol('v', positive=True) >>> from sympy import dsolve, Function >>> f = Function('f') >>> y = f(x) >>> genform = x**2*y.diff(x, 2) + x*y.diff(x) + (x**2 - v**2)*y >>> dsolve(genform) Eq(f(x), C1*besselj(v, x) + C2*bessely(v, x)) References ========== https://math24.net/bessel-differential-equation.html """ hint = "2nd_linear_bessel" has_integral = False def _matches(self): eq = self.ode_problem.eq_high_order_free f = self.ode_problem.func order = self.ode_problem.order x = self.ode_problem.sym df = f.diff(x) a = Wild('a', exclude=[f,df]) b = Wild('b', exclude=[x, f,df]) a4 = Wild('a4', exclude=[x,f,df]) b4 = Wild('b4', exclude=[x,f,df]) c4 = Wild('c4', exclude=[x,f,df]) d4 = Wild('d4', exclude=[x,f,df]) a3 = Wild('a3', exclude=[f, df, f.diff(x, 2)]) b3 = Wild('b3', exclude=[f, df, f.diff(x, 2)]) c3 = Wild('c3', exclude=[f, df, f.diff(x, 2)]) deq = a3*(f.diff(x, 2)) + b3*df + c3*f r = collect(eq, [f.diff(x, 2), df, f]).match(deq) if order == 2 and r: if not all(r[key].is_polynomial() for key in r): n, d = eq.as_numer_denom() eq = expand(n) r = collect(eq, [f.diff(x, 2), df, f]).match(deq) if r and r[a3] != 0: # leading coeff of f(x).diff(x, 2) coeff = factor(r[a3]).match(a4*(x-b)**b4) if coeff: # if coeff[b4] = 0 means constant coefficient if coeff[b4] == 0: return False point = coeff[b] else: return False if point: r[a3] = simplify(r[a3].subs(x, x+point)) r[b3] = simplify(r[b3].subs(x, x+point)) r[c3] = simplify(r[c3].subs(x, x+point)) # making a3 in the form of x**2 r[a3] = cancel(r[a3]/(coeff[a4]*(x)**(-2+coeff[b4]))) r[b3] = cancel(r[b3]/(coeff[a4]*(x)**(-2+coeff[b4]))) r[c3] = cancel(r[c3]/(coeff[a4]*(x)**(-2+coeff[b4]))) # checking if b3 is of form c*(x-b) coeff1 = factor(r[b3]).match(a4*(x)) if coeff1 is None: return False # c3 maybe of very complex form so I am simply checking (a - b) form # if yes later I will match with the standerd form of bessel in a and b # a, b are wild variable defined above. _coeff2 = r[c3].match(a - b) if _coeff2 is None: return False # matching with standerd form for c3 coeff2 = factor(_coeff2[a]).match(c4**2*(x)**(2*a4)) if coeff2 is None: return False if _coeff2[b] == 0: coeff2[d4] = 0 else: coeff2[d4] = factor(_coeff2[b]).match(d4**2)[d4] self.rn = {'n':coeff2[d4], 'a4':coeff2[c4], 'd4':coeff2[a4]} self.rn['c4'] = coeff1[a4] self.rn['b4'] = point return True return False def _get_general_solution(self, *, simplify_flag: bool = True): f = self.ode_problem.func.func x = self.ode_problem.sym n = self.rn['n'] a4 = self.rn['a4'] c4 = self.rn['c4'] d4 = self.rn['d4'] b4 = self.rn['b4'] n = sqrt(n**2 + Rational(1, 4)*(c4 - 1)**2) (C1, C2) = self.ode_problem.get_numbered_constants(num=2) return [Eq(f(x), ((x**(Rational(1-c4,2)))*(C1*besselj(n/d4,a4*x**d4/d4) + C2*bessely(n/d4,a4*x**d4/d4))).subs(x, x-b4))] class SecondLinearAiry(SingleODESolver): r""" Gives solution of the Airy differential equation .. math :: \frac{d^2y}{dx^2} + (a + b x) y(x) = 0 in terms of Airy special functions airyai and airybi. Examples ======== >>> from sympy import dsolve, Function >>> from sympy.abc import x >>> f = Function("f") >>> eq = f(x).diff(x, 2) - x*f(x) >>> dsolve(eq) Eq(f(x), C1*airyai(x) + C2*airybi(x)) """ hint = "2nd_linear_airy" has_integral = False def _matches(self): eq = self.ode_problem.eq_high_order_free f = self.ode_problem.func order = self.ode_problem.order x = self.ode_problem.sym df = f.diff(x) a4 = Wild('a4', exclude=[x,f,df]) b4 = Wild('b4', exclude=[x,f,df]) match = self.ode_problem.get_linear_coefficients(eq, f, order) does_match = False if order == 2 and match and match[2] != 0: if match[1].is_zero: self.rn = cancel(match[0]/match[2]).match(a4+b4*x) if self.rn and self.rn[b4] != 0: self.rn = {'b':self.rn[a4],'m':self.rn[b4]} does_match = True return does_match def _get_general_solution(self, *, simplify_flag: bool = True): f = self.ode_problem.func.func x = self.ode_problem.sym (C1, C2) = self.ode_problem.get_numbered_constants(num=2) b = self.rn['b'] m = self.rn['m'] if m.is_positive: arg = - b/cbrt(m)**2 - cbrt(m)*x elif m.is_negative: arg = - b/cbrt(-m)**2 + cbrt(-m)*x else: arg = - b/cbrt(-m)**2 + cbrt(-m)*x return [Eq(f(x), C1*airyai(arg) + C2*airybi(arg))] class LieGroup(SingleODESolver): r""" This hint implements the Lie group method of solving first order differential equations. The aim is to convert the given differential equation from the given coordinate system into another coordinate system where it becomes invariant under the one-parameter Lie group of translations. The converted ODE can be easily solved by quadrature. It makes use of the :py:meth:`sympy.solvers.ode.infinitesimals` function which returns the infinitesimals of the transformation. The coordinates `r` and `s` can be found by solving the following Partial Differential Equations. .. math :: \xi\frac{\partial r}{\partial x} + \eta\frac{\partial r}{\partial y} = 0 .. math :: \xi\frac{\partial s}{\partial x} + \eta\frac{\partial s}{\partial y} = 1 The differential equation becomes separable in the new coordinate system .. math :: \frac{ds}{dr} = \frac{\frac{\partial s}{\partial x} + h(x, y)\frac{\partial s}{\partial y}}{ \frac{\partial r}{\partial x} + h(x, y)\frac{\partial r}{\partial y}} After finding the solution by integration, it is then converted back to the original coordinate system by substituting `r` and `s` in terms of `x` and `y` again. Examples ======== >>> from sympy import Function, dsolve, exp, pprint >>> from sympy.abc import x >>> f = Function('f') >>> pprint(dsolve(f(x).diff(x) + 2*x*f(x) - x*exp(-x**2), f(x), ... hint='lie_group')) / 2\ 2 | x | -x f(x) = |C1 + --|*e \ 2 / References ========== - Solving differential equations by Symmetry Groups, John Starrett, pp. 1 - pp. 14 """ hint = "lie_group" has_integral = False def _has_additional_params(self): return 'xi' in self.ode_problem.params and 'eta' in self.ode_problem.params def _matches(self): eq = self.ode_problem.eq f = self.ode_problem.func.func order = self.ode_problem.order x = self.ode_problem.sym df = f(x).diff(x) y = Dummy('y') d = Wild('d', exclude=[df, f(x).diff(x, 2)]) e = Wild('e', exclude=[df]) does_match = False if self._has_additional_params() and order == 1: xi = self.ode_problem.params['xi'] eta = self.ode_problem.params['eta'] self.r3 = {'xi': xi, 'eta': eta} 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) self.r3.update(r) does_match = True return does_match def _get_general_solution(self, *, simplify_flag: bool = True): eq = self.ode_problem.eq x = self.ode_problem.sym func = self.ode_problem.func order = self.ode_problem.order df = func.diff(x) try: eqsol = solve(eq, df) except NotImplementedError: eqsol = [] desols = [] for s in eqsol: sol = _ode_lie_group(s, func, order, match=self.r3) if sol: desols.extend(sol) if desols == []: raise NotImplementedError("The given ODE " + str(eq) + " cannot be solved by" + " the lie group method") return desols solver_map = { 'factorable': Factorable, 'nth_linear_constant_coeff_homogeneous': NthLinearConstantCoeffHomogeneous, 'nth_linear_euler_eq_homogeneous': NthLinearEulerEqHomogeneous, 'nth_linear_constant_coeff_undetermined_coefficients': NthLinearConstantCoeffUndeterminedCoefficients, 'nth_linear_euler_eq_nonhomogeneous_undetermined_coefficients': NthLinearEulerEqNonhomogeneousUndeterminedCoefficients, 'separable': Separable, '1st_exact': FirstExact, '1st_linear': FirstLinear, 'Bernoulli': Bernoulli, 'Riccati_special_minus2': RiccatiSpecial, '1st_rational_riccati': RationalRiccati, '1st_homogeneous_coeff_best': HomogeneousCoeffBest, '1st_homogeneous_coeff_subs_indep_div_dep': HomogeneousCoeffSubsIndepDivDep, '1st_homogeneous_coeff_subs_dep_div_indep': HomogeneousCoeffSubsDepDivIndep, 'almost_linear': AlmostLinear, 'linear_coefficients': LinearCoefficients, 'separable_reduced': SeparableReduced, 'nth_linear_constant_coeff_variation_of_parameters': NthLinearConstantCoeffVariationOfParameters, 'nth_linear_euler_eq_nonhomogeneous_variation_of_parameters': NthLinearEulerEqNonhomogeneousVariationOfParameters, 'Liouville': Liouville, '2nd_linear_airy': SecondLinearAiry, '2nd_linear_bessel': SecondLinearBessel, '2nd_hypergeometric': SecondHypergeometric, 'nth_order_reducible': NthOrderReducible, '2nd_nonlinear_autonomous_conserved': SecondNonlinearAutonomousConserved, 'nth_algebraic': NthAlgebraic, 'lie_group': LieGroup, } # Avoid circular import: from .ode import dsolve, ode_sol_simplicity, odesimp, homogeneous_order