# PrimitiveRatDE(F, UP, L, LQ)¶

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

`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`

.