Curvature-dependent Riccati connection

For the surface of a first-order ODE $u^{'} = ϕ (x, u)$ , the divergence of $A = \partial_{x} + ϕ \partial_{u}$ is $div A = ϕ_{u}$ . Evaluating along a solution $f$ gives $p_{f} (x) = ϕ_{u} (x, f (x))$ .

Theorem. The curvature satisfies $K (x, u) = κ (x)$ , i.e., it is a u-independent curvature ODEs, iff for every solution $f$ , $p_{f}$ satisfies the Riccati equation

p^{'} + p^{2} + κ (x) = 0.

Proof. From $K = - A (ϕ_{u}) - ϕ_{u}^{2}$ , evaluating along a solution gives $p_{f}^{'} + p_{f}^{2} + K (x, f (x)) = 0$ .

The substitution $p = y^{'} / y$ linearizes this to the Schrödinger-type equation

y^{″} + κ (x) y = 0,

establishing a direct link between the curvature condition and the linear operator $L = d^{2} / d x^{2} + κ (x)$ .

Reference: @panalvarezcurvature