The classical quantization of a family of a quadratic Liénard-type equation (Liénard II equation) is achieved by a quantization scheme (Nucci 2011)  that preserves the Noether point symmetries of the underlying Lagrangian in order to construct the Schrödinger equation. This method straightforwardly yields the Schrödinger equation as given in Choudhury and Guha (2013) .
Gubbiotti, G., Nucci, M.C. (2015). Quantization of quadratic Liénard-type equations by preserving Noether symmetries. JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, 422(2), 1235-1246 [10.1016/j.jmaa.2014.09.045].