First-order ODEs with curvature K(x,u)=κ(x) are integrable by quadratures exactly when the associated linear operator L=d²/dx²+κ(x) has a non-zero Liouvillian solution, with Kovacic's algorithm deciding the rational-κ case.
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From curvature to Kovacic: a geometric approach to integrability of scalar ODEs
First-order ODEs with curvature K(x,u)=κ(x) are integrable by quadratures exactly when the associated linear operator L=d²/dx²+κ(x) has a non-zero Liouvillian solution, with Kovacic's algorithm deciding the rational-κ case.