ODE embedding into linear equations

For first-order ODEs $u^{\prime }=\varphi (x,u)$ whose Gauss curvature of an ODE-surface satisfies $\mathcal{K}(x,u)=\kappa (x)$, that is, u-independent curvature ODEs, every solution $u$ of the nonlinear ODE also satisfies a fixed non-homogeneous linear equation.

Theorem. If $\mathcal{K}(x,u)=\kappa (x)$, there exists $c(x)$ such that every local solution $u$ satisfies

Conversely, if there exist $k(x),c(x)$ such that every solution satisfies $u^{″}+k(x)u=c(x)$, then $\mathcal{K}(x,u)=k(x)$.

Characterization. A first-order ODE satisfies $\mathcal{K}(x,u)=\kappa (x)$ iff its solution set $\mathrm{\Gamma }$ is contained in the affine space $S_L=v+\ker L$, where $L=d^2/d{x}^2+\kappa (x)$.

This embedding, together with the curvature-dependent Riccati connection, links the nonlinear ODE to the linear operator $L$, making differential Galois theory applicable to decide integrability.

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

Reference: @panalvarezcurvature