RationalRicDE(F, UP)

riccati.spad line 254 [edit on github]

• 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)), Mapping(List(F), UP)) -> 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)) -> List(Fraction(UP)) if F has AlgebraicallyClosedField

ricDsolve(op) returns the rational solutions of the associated Riccati equation of op y = 0.

ricDsolve : (LinearOrdinaryDifferentialOperator1(Fraction(UP)), Mapping(Factored(UP), 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)), Mapping(List(F), UP)) -> 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)), Mapping(List(F), UP), Mapping(Factored(UP), 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)) -> 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)), Mapping(Factored(UP), 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)), Mapping(List(F), UP)) -> 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)), Mapping(List(F), UP), Mapping(Factored(UP), 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.

singRicDE : (LinearOrdinaryDifferentialOperator2(UP, Fraction(UP)), Mapping(Factored(UP), 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.