# RationalLODE(F, UP)¶

RationalLODE provides functions for in-field solutions of linear ordinary differential equations, in the rational case.

indicialEquationAtInfinity: LinearOrdinaryDifferentialOperator1 Fraction UP -> UP
indicialEquationAtInfinity op returns the indicial equation of op at infinity.
indicialEquationAtInfinity: LinearOrdinaryDifferentialOperator2(UP, Fraction UP) -> UP
indicialEquationAtInfinity op returns the indicial equation of op at infinity.
integrate_sols: LinearOrdinaryDifferentialOperator1 Fraction UP -> Record(ltilde: LinearOrdinaryDifferentialOperator1 Fraction UP, r: Union(LinearOrdinaryDifferentialOperator1 Fraction UP, failed))
integrate_sols(l) integrates the solutions of an operator l.
ratDsolve: (LinearOrdinaryDifferentialOperator1 Fraction UP, Fraction UP) -> Record(particular: Union(Fraction UP, failed), basis: List Fraction UP)
ratDsolve(op, g) returns ["failed", []] if the equation op y = g has no rational solution. Otherwise, it returns [f, [y1, ..., ym]] where f is a particular rational solution and the yi's form a basis for the rational solutions of the homogeneous equation.
ratDsolve: (LinearOrdinaryDifferentialOperator1 Fraction UP, List Fraction UP) -> Record(basis: List Fraction UP, mat: Matrix F)
ratDsolve(op, [g1, ..., gm]) returns [[h1, ..., hq], M] such that any rational solution of op y = c1 g1 + ... + cm gm is of the form c1 h1 + ... + cq hq where M [c1, ..., cq] = 0 and q >= m.
ratDsolve: (LinearOrdinaryDifferentialOperator2(UP, Fraction UP), Fraction UP) -> Record(particular: Union(Fraction UP, failed), basis: List Fraction UP)
ratDsolve(op, g) returns ["failed", []] if the equation op y = g has no rational solution. Otherwise, it returns [f, [y1, ..., ym]] where f is a particular rational solution and the yi's form a basis for the rational solutions of the homogeneous equation.
ratDsolve: (LinearOrdinaryDifferentialOperator2(UP, Fraction UP), List Fraction UP) -> Record(basis: List Fraction UP, mat: Matrix F)
ratDsolve(op, [g1, ..., gm]) returns [[h1, ..., hq], M] such that any rational solution of op y = c1 g1 + ... + cm gm is of the form c1 h1 + ... + cq hq where M [c1, ..., cq] = 0 and q >= m.