Differential Galois theory

Differential Galois theory (Picard–Vessiot theory) studies the algebraic structure of solutions to linear ordinary differential equations. For a linear ODE

L (y) = y^{(n)} + a_{n - 1} (x) y^{(n - 1)} + \dots + a_{0} (x) y = 0,

the differential Galois group $G$ is the group of field automorphisms of the Picard–Vessiot extension that commute with differentiation and leave the coefficient field fixed.

A solution is Liouvillian if it can be expressed in terms of exponentials, integrals, and algebraic functions of known functions. The key result: $L (y) = 0$ has a non-zero Liouvillian solution iff the identity component of $G$ is solvable.

For second-order linear ODEs $y^{″} + r (x) y = 0$ with $r \in C (x)$ , the Kovacic algorithm decides Liouvillian solvability.