Quantization of the Liénard II equation and Jacobi's last multiplier

In this paper, the role of Jacobi's last multiplier in mechanical systems with a position-dependent mass is unveiled. In particular, we map the Liénard II equation x + f (x) x² + g(x) = 0 to a position-dependent mass system. The quantization of the Liénard II equation is then carried out using the point canonical transformation method together with the Von Roos ordering technique. Finally, we show how their eigenfunctions and eigenspectrum can be obtained in terms of associated Laguerre and exceptional Laguerre functions.