On the integrability in finite terms of first-order differential equations according to Maximovic

Abstract

We give a definition of integration by quadratures of first-order ordinary differential equations, and recover a little known result by Maximovic which states that a first-order ordinary differential equation can be integrated by quadratures only if it arises from the linear equation by a diffeomorphic transformation of the dependent variable. In the appendix this result is applied to the linear second-order equation in a restricted setting. A brief outline of Maximovic's life is also included.