Classical and relativistic laws of motion for spherical supernovas

We derive some first order differential equations which model the classical and the relativistic thin layer approximations. The circumstellar medium is assumed to follow a density profile which can be exponential, Gaussian, Plummer-like, self-gravitating of Lane–Emden ($n=5$) type, or power law. The first order differential equations are solved analytically, or numerically, or by a series expansion, or by recursion, or by Pad'e approximation. The initial conditions are chosen in order to model the temporal evolution of SN 1993J over ten years. The Pad'e approximated equations of motion are applied to four SNRs: Tycho, Cas A, Cygnus loop, and SN~1006.