Kovacic algorithm

The Kovacic algorithm decides whether the second-order linear ODE

admits a non-zero Liouvillian solution (see differential Galois theory), and constructs it when it exists.

It proceeds by analyzing the poles of $r(x)$ at finite points and at infinity, classifying into three cases:

1. Case 1: a solution of the form $e^{\int \omega dx}$ with $\omega \in \mathbb{C}(x)$ (rational)
2. Case 2: a solution where $\omega$ is algebraic of degree 2 over $\mathbb{C}(x)$
3. Case 3: a solution where $\omega$ is algebraic of degree 4, 6, or 12

If none of the three cases applies, no Liouvillian solution exists.

Applied to the Galois criterion for ODE integrability, it gives an effective procedure to decide whether a first-order ODE with $\mathcal{K}(x,u)=\kappa (x)\in \mathbb{C}(x)$ is integrable by quadratures.

Related notes: differential Galois theory, Galois criterion for ODE integrability, curvature-dependent Riccati connection