# Kovacic(F, UP)¶

kovacic.spad line 1 [edit on github]

F: Join(CharacteristicZero, AlgebraicallyClosedField, RetractableTo Integer, RetractableTo Fraction Integer)

Kovacic provides a modified Kovacic`'s`

algorithm for solving explicitly irreducible 2nd order linear ordinary differential equations.

- kovacic: (Fraction UP, Fraction UP, Fraction UP) -> Union(SparseUnivariatePolynomial Fraction UP, failed)
`kovacic(a_0,a_1,a_2)`

returns either “failed” or`P`

(`u`

) such that`\$e^{\int(-a_1/2a_2)} e^{\int u}\$`

is a solution of`a_2 y'' + a_1 y' + a0 y = 0`

whenever`u`

is a solution of`P u = 0`

. The equation must be already irreducible over the rational functions.

- kovacic: (Fraction UP, Fraction UP, Fraction UP, UP -> Factored UP) -> Union(SparseUnivariatePolynomial Fraction UP, failed)
`kovacic(a_0,a_1,a_2,ezfactor)`

returns either “failed” or`P`

(`u`

) such that`\$e^{\int(-a_1/2a_2)} e^{\int u}\$`

is a solution of`\$a_2 y'' + a_1 y' + a0 y = 0\$`

whenever`u`

is a solution of`P u = 0`

. The equation must be already irreducible over the rational functions. Argument`ezfactor`

is a factorisation in`UP`

, not necessarily into irreducibles.