Einstein equations for several matter sources in Robertson-Walker and Bianchi
I type metrics, are shown to reduce to a kind of second order nonlinear
ordinary differential equation y¨+αf(y)y˙+βf(y)∫f(y)dy+γf(y)=0. Also, it appears in the generalized statistical mechanics
for the most interesting value q=-1. The invariant form of this equation is
imposed and the corresponding nonlocal transformation is obtained. The
linearization of that equation for any α,β and γ is
presented and for the important case f=byn+k with β=α2(n+1)/((n+2)2) its explicit general solution is found. Moreover, the form
invariance is applied to yield exact solutions of same other differential
equations.Comment: 22 pages, RevTeX; to appear in J. Math. Phy