# RationalRicDE(F, UP)¶

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.