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

Classical and relativistic laws of motion for spherical supernovas