On the connection of the quadratic Liénard equation with an equation for the elliptic functions (Q892769)

Consider the differential equation \[ {d^2y\over dx^2}+ g(y)\Biggl({dy\over dx}\Biggr)^2+ h(y)= 0.\tag{\(*\)} \] The authors prove that \((*)\) can be transformed into the differential equation \[ w{d^2w\over d\xi^2}-{1\over 2}\Biggl({dw\over d\xi}\Biggr)^2+ 4\omega^3=0\tag{\(**\)} \] by means of the nonlocal transformation \[ w=F(y),\quad d\xi= G(y)\,dx. \] It is important that the general solution of \((**)\) has the form \[ w=-\wp(\xi-\xi_0,c,0). \] By this way it is possible under some conditions to present an explicit expression for the general solution of \((*)\).