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We use the solution space of a pair of ODEs of at least second order toconstruct a smooth surface in Euclidean space. We describe when this surface isa proper embedding which is geodesically complete with finite total Gausscurvature. If the associated roots of the ODEs are real and distinct, we give auniversal upper bound for the total Gauss curvature of the surface whichdepends only on the orders of the ODEs and we show that the total Gausscurvature of the surface vanishes if the ODEs are second order. We examine whenthe surfaces are asymptotically minimal.

Real analytic complete non-compact surfaces in Euclidean space with finite total curvature arising as solutions to ODEs