A powerful diagnostic tool of analytic solutions of ordinary second-order linear homogeneous differential equations

Very few of the ordinary second-order linear homogeneous (OSLH) differential equations are known to possess analytic, closed-form solutions and there is no general theory to predict whether such solutions exist in each particular case. In this work, we present a powerful new method for the discovery of closed-form solutions in the entire class of the OSLH differential equations. A sufficient condition for the existence of such solutions is that the equations can be transformed to forms with constant coefficients and we show that two predictors exist, a nonlinear first-order Bernoulli equation with index n=3/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$n=3/2$\end{document} and a nonlinear second-order equation. There is no need to solve these equations, only to verify them based on the contents of a given OSLH equation. Because the predictors can be verified quite easily in all cases, we believe that this methodology is an important new diagnostic tool for studying all the equations of mathematical physics that belong to the OSLH class.