2 2 for more discussion of the frobenius method as

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Unformatted text preview: 2 → 0 as x → 1 and hence the quotients 2x 1 − x2 p(p + 1) 1 − x2 and blow up as x → 1, but it is a regular singular point because (x − 1)P (x) = (x − 1) and (x − 1)2 Q(x) = (x − 1)2 −2x 2x = 2 1−x x+1 p(p + 1) p(p + 1)(1 − x) = 1 − x2 1+x are both real analytic at x0 = 1. The point of these definitions is that in the case where x = x0 is a regular singular point, a modification of the power series method can still be used to find solutions. Theorem of Frobenius. If x0 is a regular singular point for the differential equation d2 y dy + P (x) + Q(x)y = 0, 2 dx dx then this differential equation has at least one nonzero solution of the form ∞ y (x) = (x − x0 )r an (x − x0 )n , (1.14) n=0 where r is a constant, which may be complex. If (x − x0 )P (x) and (x − x0 )2 Q(x) have power series which converge for |x − x0 | < R then the power series ∞ an (x − x0 )n n=0 will also converge for |x − x0 | < R. We will call a solution of the form (1.14) a generalized power series solutio...
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This document was uploaded on 01/12/2014.

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