Flatness-integrability theorem

Theorem. If the Gaussian curvature $\mathcal{K}$ of the surface associated to a first-order ODE $u^{\prime }=\varphi (x,u)$ is zero, then the ODE can be fully integrated by quadratures.

Proof. Flatness $\mathcal{K}=0$ gives ${\partial }_u(A(\varphi ))=0$, so $A(\varphi )$ depends only on $x$. Choose ${\mathrm{\Psi }}^{\prime }(x)=A(\varphi )$. Then $F(x,u)=\varphi (x,u)-\mathrm{\Psi }(x)$ is a first integral of $A$, and the general solution is $\varphi (x,u)-\mathrm{\Psi }(x)=C$.

This is the $k=0$ instance of the constant curvature integrability theorem.

Related notes: constant curvature integrability, relative Jacobi field, Gauss curvature of an ODE-surface, Riemannian metric for a first-order ODE

Reference: @panalvarez2023surfaces