Section A.6 The Method of FrobeniusA197Solutions to Exercises A.61.For the equationy±±+(1-x2)y±+xy=0,p(x)=1-x2andq(x)=xare bothanalytic ata= 0. Soa= 0 is a ordinary point.5.For the equationx2y±±-ex)y±+xy=0,p(x1-exx2,xp(x1-exx;q(x1x2q(xx.p(x) andq(x) are not analytic at 0. Soa= 0 is a singular point. Sincexp(x) andx2q(x) are analytic ata= 0, the pointa= 0 is a regular singular point. To seethatxp(x) is analytic at 0, derive its Taylor series as follows: for allx,ex=1+x+x22!+x33!+···1-ex=-x-x22!-x33!+=x±-1-x2!-x23!+²1-exx=-1-x2!-x23!+.Since1-exxhas a Taylor series expansion about 0 (valid for allx), it is analytic at0.9.For the equation 4x2y±±-14xy±+ (20-x)y,p(x-72xp(x-72,p0=-72;q(x20-x4x22q(x)=5-x4,q0=5.p(x) andq(x) are not analytic at 0. Soa= 0 is a singular point. Sincexp(x) andx2q(x) are analytic ata= 0, the point
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