# RationalLODE(F, UP)ΒΆ

- F: Join(Field, CharacteristicZero, RetractableTo Integer, RetractableTo Fraction Integer)
- UP: UnivariatePolynomialCategory F

`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`

.