A generalized Cole–Hopf transformation for nonlinear ODEs

We introduce a 'hybrid' Cole–Hopf–Darboux transformation to relate the solutions of nonlinear and linear second-order differential equations and derive classification and sufficient condition for this correspondence. We explore physical applications of this correspondence to nonlinear oscillations, the Duffing equation and a nonlinear form of the Schrödinger equation for the nonrelativistic hydrogen atom. In addition, we show that solutions of some nonlinear second-order equations are related to the special functions of mathematical physics through this transformation. These nonlinear equations can be viewed as the 'class of special nonlinear equations' which correspond to the linear differential equations which define the special functions of mathematical physics. (paper)