# 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.