Minimal Surfaces in Three-Dimensional Riemannian Manifold Associated with a Second-Order ODE

We show that a surface corresponding to a first-order ODE is minimal in three-dimensional Riemannian manifold which is determined by the first prolongation of ({text {d}}y/mathrm{d}x=p(x,y)) if and only if (p_{yy}=0). Accordingly, any linear first-order ODE describes a minimal surface which is not necessarily totally geodesic.