Form Invariance of Differential Equations in General Relativity

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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.