A generalized Cole-Hopf transformation for nonlinear ODEs (Q2847977)

As is well known, the nonlinear Riccati equation can be transformed into a second-order linear equation by a special case of the Cole-Hopf transformation. Analogously, a generalized Cole-Hopf transformation is used to transform a second-order ordinary differential equation (ODE) with cubic nonlinearities into a second-order linear ODE, and vice versa. A sufficient condition for this correspondence is derived. The method is applied to some important equations of mathematical physics, such as the Schrödinger equation, the Duffing equation, and the radial equation for the hydrogen atom. Since many special functions of mathematical physics, such as the Legendre polynomials, Bessel functions, Laguerre and Hermite polynomials, satisfy linear second-order ODEs, the correspondence established in this paper also yields a special class of nonlinear equations whose solutions can be expressed in terms of the above-mentioned special functions.