Form Invariance of Differential Equations in General Relativity

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 $\ddot{y}+\alpha f(y)\dot{y}+\beta f(y)\int{f(y) dy}+\gamma f(y)=0$. 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 $\alpha, \beta$ and $\gamma$ is presented and for the important case $f=by^n+k$ with $\beta=\alpha ^2 (n+1)/((n+2)^2)$ its explicit general solution is found. Moreover, the form invariance is applied to yield exact solutions of same other differential equations.