For example to solve the dierential equation x2 d2 y

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Unformatted text preview: n . Unfortunately, the theorem guarantees only one generalized power series solution, not a basis. In fortuitous cases, one can find a basis of generalized power series solutions, but not always. The method of finding generalized power series solutions to (1.13) in the case of regular singular points is called the Frobenius method .2 2 For more discussion of the Frobenius method as well as many of the other techniques touched upon in this chapter we refer the reader to George F. Simmons, Differential equations with applications and historical notes , second edition, McGraw-Hill, New York, 1991. 16 The simplest differential equation to which the Theorem of Frobenius applies is the Cauchy-Euler equidimensional equation. This is the special case of (1.13) for which p q P (x) = , Q(x) = 2 , x x where p and q are constants. Note that xP (x) = p x2 Q(x) = q and are real analytic, so x = 0 is a regular singular point for the Cauchy-Euler equation as long as either p or q is nonzero. The Frobenius method...
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