# PrimitiveRatDE(F, UP, L, LQ)¶

`PrimitiveRatDE` provides functions for in-field solutions of linear ordinary differential equations, in the transcendental case. The derivation to use is given by the parameter `L`.

denomLODE: (L, Fraction UP) -> Union(UP, failed)

`denomLODE(op, g)` returns a polynomial `d` such that any rational solution of `op y = g` is of the form `p/d` for some polynomial `p`, and “failed”, if the equation has no rational solution.

denomLODE: (L, List Fraction UP) -> UP

`denomLODE(op, [g1, ..., gm])` returns a polynomial `d` such that any rational solution of `op y = c1 g1 + ... + cm gm` is of the form `p/d` for some polynomial `p`.

indicialEquation: (L, F) -> UP

`indicialEquation(op, a)` returns the indicial equation of `op` at `a`.

indicialEquation: (LQ, F) -> UP

`indicialEquation(op, a)` returns the indicial equation of `op` at `a`.

indicialEquations: (L, UP) -> List Record(center: UP, equation: UP)

`indicialEquations(op, p)` returns `[[d1, e1], ..., [dq, eq]]` where the `d_i``'s` are the affine singularities of `op` above the roots of `p`, and the `e_i``'s` are the indicial equations at each `d_i`.

indicialEquations: (LQ, UP) -> List Record(center: UP, equation: UP)

`indicialEquations(op, p)` returns `[[d1, e1], ..., [dq, eq]]` where the `d_i``'s` are the affine singularities of `op` above the roots of `p`, and the `e_i``'s` are the indicial equations at each `d_i`.

indicialEquations: L -> List Record(center: UP, equation: UP)

`indicialEquations op` returns `[[d1, e1], ..., [dq, eq]]` where the `d_i``'s` are the affine singularities of `op`, and the `e_i``'s` are the indicial equations at each `d_i`.

indicialEquations: LQ -> List Record(center: UP, equation: UP)

`indicialEquations op` returns `[[d1, e1], ..., [dq, eq]]` where the `d_i``'s` are the affine singularities of `op`, and the `e_i``'s` are the indicial equations at each `d_i`.

splitDenominator: (LQ, List Fraction UP) -> Record(eq: L, rh: List Fraction UP)

`splitDenominator(op, [g1, ..., gm])` returns `op0, [h1, ..., hm]` such that the equations `op y = c1 g1 + ... + cm gm` and `op0 y = c1 h1 + ... + cm hm` have the same solutions.