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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
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)
P (x) = ,
Q(x) = 2 ,
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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- Winter '14