Can someone please show the EFE solved for the entire universe. I've heard people say that 'Einstein solved the Einstein equations for the universe and found this'. If I am missing something, what does this statement really imply?
"Equations for the whole universe" were not solved by Einstein. The Universe with her every captivating quirk and je ne sais quoi is impossible to fathom. The only persons who claimed ultimate solution are the fortune tellers and the Wall Street pundits who purportedly can predict tomorrow's SNP 500.
However, when it comes to the coarse-grained approximate solution (assuming that She is homogeneous and isotropic), there are at least four guys (Friedmann–Lemaître–Robertson–Walker) who beat Einstein to the punch. See wiki quotes below:
The Soviet mathematician Alexander Friedmann first derived the main results of the FLRW model in 1922 and 1924. Although the prestigious physics journal Zeitschrift für Physik published his work, it remained relatively unnoticed by his contemporaries. Friedmann was in direct communication with Albert Einstein, who, on behalf of Zeitschrift für Physik, acted as the scientific referee of Friedmann's work. Eventually Einstein acknowledged the correctness of Friedmann's calculations, but failed to appreciate the physical significance of Friedmann's predictions.
Friedmann died in 1925. In 1927, Georges Lemaître, a Belgian priest, astronomer and periodic professor of physics at the Catholic University of Leuven, arrived independently at results similar to those of Friedmann and published them in the Annales de la Société Scientifique de Bruxelles (Annals of the Scientific Society of Brussels). In the face of the observational evidence for the expansion of the universe obtained by Edwin Hubble in the late 1920s, Lemaître's results were noticed in particular by Arthur Eddington, and in 1930–31 Lemaître's paper was translated into English and published in the Monthly Notices of the Royal Astronomical Society.
Howard P. Robertson from the US and Arthur Geoffrey Walker from the UK explored the problem further during the 1930s. In 1935 Robertson and Walker rigorously proved that the FLRW metric is the only one on a spacetime that is spatially homogeneous and isotropic (as noted above, this is a geometric result and is not tied specifically to the equations of general relativity, which were always assumed by Friedmann and Lemaître).
What Einstein did is tweaking (adding and subsequently dropping a constant term) the equation to make the solution palatable.
To make it worse, Einstein was not even the first to write down the correct version of Einstein equation. It's the great mathematician Hilbert who proposed the action of gravity in 1915, prior to the celebrated 1916 paper of Einstein.