Abstract
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 ÿ+αf(y)ẏ+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 some other differential equations.
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