Mathematical Methods for Physics and Engineering - Matematica.NET

PDES: SEPARATION OF VARIABLES AND OTHER METHODSThis can be rearranged to giver 2 R 1 ′′ +2rR 1 ′ − 2R 1 = − Ar3 ,ɛ 0where the prime denotes differentiation with respect to r. The LHS is homogeneous andthe equation can be reduced by the substitution r =expt, and writing R 1 (r) =S(t), to¨S + Ṡ − 2S = − A ɛ 0exp 3t, (21.68)where the dots indicate differentiation with respect to t.This is an inhomogeneous second-order ODE with constant coefficients and can bestraightforwardly solved by the methods of subsection 15.2.1 to giveRecalling that r =expt we findS(t) =c 1 exp t + c 2 exp(−2t) −A10ɛ 0exp 3t.R 1 (r) =c 1 r + c 2 r −2 −A r 3 .10ɛ 0Since we are interested in the region r