On the integrability of differential equations by quadratures according to Maximovič

We show that a first-order ordinary differential equation can be integrated by quadratures in the sense of Maximovič only if it arises from the linear equation by a diffeomorphic transformation of the dependent variable. In the appendix this result is applied to show that the linear second-order equation can be integrated by quadratures in a restricted sense only if it has constant coefficients. A brief outline of Maximovičʹs life is also included.