# 2x one easily checks that x 0 is a regular singular

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Unformatted text preview: and all the other an ’s and bn ’s are zero, so both of these solutions are generalized power series solutions. 17 On the other hand, if this method is applied to the diﬀerential equation x2 d2 y dy + y = 0, + 3x 2 dx dx we obtain r(r − 1) + 3r + 1 = r2 + 2r + 1, which has a repeated root. In this case, we obtain only a one-parameter family of solutions y = cx−1 . Fortunately, there is a trick that enables us to handle this situation, the so-called method of variation of parameters. In this context, we replace the parameter c by a variable v (x) and write y = v (x)x−1 . Then d2 y = v (x)x−1 − 2v (x)x−2 + 2v (x)x−3 . dx2 dy = v (x)x−1 − v (x)x−2 , dx Substitution into the diﬀerential equation yields x2 (v (x)x−1 − 2v (x)x−2 + 2v (x)x−3 ) + 3x(v (x)x−1 − v (x)x−2 ) + v (x)x−1 = 0, which quickly simpliﬁes to yield xv (x) + v (x) = 0, v 1 =− , v x log |v | = − log |x| + a, v= c2 , x where a and c2 are constants of integration. A further integration yields v = c2 log |x| + c1 , y = (c2 log |x|...
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## This document was uploaded on 01/12/2014.

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