# RationalRicDE(F, UP)ΒΆ

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

In-field solution of Riccati equations, rational case.

- polyRicDE: (LinearOrdinaryDifferentialOperator2(UP, Fraction UP), UP -> List F) -> List Record(poly: UP, eq: LinearOrdinaryDifferentialOperator2(UP, Fraction UP))
`polyRicDE(op, zeros)`

returns`[[p1, L1], [p2, L2], ... , [pk, Lk]]`

such that the polynomial part of any rational solution of the associated Riccati equation of`op y = 0`

must be one of the`pi`

`'s`

(up to the constant coefficient), in which case the equation for`z = y e^{-int p}`

is`Li z = 0`

.`zeros`

is a zero finder in`UP`

.

- ricDsolve: (LinearOrdinaryDifferentialOperator1 Fraction UP, UP -> Factored UP) -> List Fraction UP if F has AlgebraicallyClosedField
`ricDsolve(op, ezfactor)`

returns the rational solutions of the associated Riccati equation of`op y = 0`

. Argument`ezfactor`

is a factorisation in`UP`

, not necessarily into irreducibles.

- ricDsolve: (LinearOrdinaryDifferentialOperator1 Fraction UP, UP -> List F) -> List Fraction UP
`ricDsolve(op, zeros)`

returns the rational solutions of the associated Riccati equation of`op y = 0`

.`zeros`

is a zero finder in`UP`

.

- ricDsolve: (LinearOrdinaryDifferentialOperator1 Fraction UP, UP -> List F, UP -> Factored UP) -> List Fraction UP
`ricDsolve(op, zeros, ezfactor)`

returns the rational solutions of the associated Riccati equation of`op y = 0`

.`zeros`

is a zero finder in`UP`

. Argument`ezfactor`

is a factorisation in`UP`

, not necessarily into irreducibles.

- ricDsolve: (LinearOrdinaryDifferentialOperator2(UP, Fraction UP), UP -> Factored UP) -> List Fraction UP if F has AlgebraicallyClosedField
`ricDsolve(op, ezfactor)`

returns the rational solutions of the associated Riccati equation of`op y = 0`

. Argument`ezfactor`

is a factorisation in`UP`

, not necessarily into irreducibles.

- ricDsolve: (LinearOrdinaryDifferentialOperator2(UP, Fraction UP), UP -> List F) -> List Fraction UP
`ricDsolve(op, zeros)`

returns the rational solutions of the associated Riccati equation of`op y = 0`

.`zeros`

is a zero finder in`UP`

.

- ricDsolve: (LinearOrdinaryDifferentialOperator2(UP, Fraction UP), UP -> List F, UP -> Factored UP) -> List Fraction UP
`ricDsolve(op, zeros, ezfactor)`

returns the rational solutions of the associated Riccati equation of`op y = 0`

.`zeros`

is a zero finder in`UP`

. Argument`ezfactor`

is a factorisation in`UP`

, not necessarily into irreducibles.

- ricDsolve: LinearOrdinaryDifferentialOperator1 Fraction UP -> List Fraction UP if F has AlgebraicallyClosedField
`ricDsolve(op)`

returns the rational solutions of the associated Riccati equation of`op y = 0`

.

- ricDsolve: LinearOrdinaryDifferentialOperator2(UP, Fraction UP) -> List Fraction UP if F has AlgebraicallyClosedField
`ricDsolve(op)`

returns the rational solutions of the associated Riccati equation of`op y = 0`

.

- singRicDE: (LinearOrdinaryDifferentialOperator2(UP, Fraction UP), UP -> Factored UP) -> List Record(frac: Fraction UP, eq: LinearOrdinaryDifferentialOperator2(UP, Fraction UP))
`singRicDE(op, ezfactor)`

returns`[[f1, L1], [f2, L2], ..., [fk, Lk]]`

such that the singular`++`

part of any rational solution of the associated Riccati equation of`op y = 0`

must be one of the`fi`

`'s`

(up to the constant coefficient), in which case the equation for`z = y e^{-int ai}`

is`Li z = 0`

. Argument`ezfactor`

is a factorisation in`UP`

, not necessarily into irreducibles.