Galois criterion for ODE integrability

For first-order ODEs $u^{\prime }=\varphi (x,u)$ with curvature $\mathcal{K}(x,u)=\kappa (x)$ (u-independent curvature ODEs), the ODE embedding into linear equations gives $L=d^2/d{x}^2+\kappa (x)$.

Theorem. The nonlinear ODE is integrable by quadratures iff $L$ admits a non-zero Liouvillian solution (see differential Galois theory).

When $\kappa \in \mathbb{C}(x)$, the Kovacic algorithm provides an effective decision procedure, making Liouvillian integrability algorithmically decidable for this class.

This bridges two theories:

• The geometric surface framework (curvature → Riccati → Schrödinger)
• differential Galois theory of the linear operator $L$

The approach complements Prelle–Singer methods: rather than searching directly in the nonlinear equation, it reduces to the Galois theory of $L$.

Related notes: curvature-dependent Riccati connection, ODE embedding into linear equations, differential Galois theory, Kovacic algorithm

Reference: @panalvarezcurvature