First integrals for the modified Emden equation <i>q</i>+α(<i>t</i>) <i>q</i>+<i>q</i> <i>n</i> =0

Abstract

It is shown that the modified Emden equation q̈+α(t)q̇+qn=0 possesses first integrals for functions α(t) other than kt−1. The function α(t) is obtained explicitly in the case n=3 and parametrically for other n(≠2). The case n=2 is seen to be particularly difficult to solve.