Equivalence classes for Emden equations (Q1612592)

Emden equations of the type \[ y''(t)+ e(t) y^p(t)= 0\tag{1} \] are considered. The authors call two second-order differential equations \[ (2)\qquad y''= F(t,y,y')\quad\text{and}\qquad (3)\qquad \eta''= f(\tau,\eta,\eta') \] to be equivalent if there exists a transformation \(t= \varphi(\tau,\eta)\), \(y= \psi(\tau,\eta)\) (\(\varphi\), \(\psi\) are smooth functions, \(\tau\) is a new independent variable and \(y\) is a new function) which transforms (2) and (3) into each other. For special \(F\) and \(f\), sufficiently conditions for the equivalence between (2) and (3) are given.