622 CHAPTER 32. SOLUTIONS NEAR A REGULAR SINGULAR POINT

√2

−2+√

2[4−4

√2][

16−8√

2]x4 + · · ·

Then the general solution is

C1y1 +C2y2

and this is valid for x > 0. Note that the ratio of the two solutions is not a constant so thisis indeed the general solution.

Generalities

For an equationx2y′′+ xp(x)y′+q(x)y = 0

having a regular singular point at 0, one looks for solutions in the form

y(x) =∞

∑n=0

anxr+n (32.12)

where r is a constant which is to be determined, in such a way that a0 ̸= 0. It turns out thatsuch equations always have such solutions although solutions of this sort are not alwaysenough to obtain the general solution to the equation. The constant r is called the exponentof the singularity because the solution is of the form

xra0 + higher order terms.

Thus the behavior of the solution to the equation given above is like xr for x near thesingularity, 0.

If you require that 32.12 solves 32.9 and plug in, you obtain using Theorem 32.2.3

∞

∑n=0

(r+n)(r+n−1)anxn+r ++∞

∑n=0

(n

∑k=0

ak (k+ r)bn−k

)xn+r

+∞

∑n=0

(n

∑k=0

cn−kak

)xn+r = 0. (32.13)

Since a0 ̸= 0,p(r)≡ r (r−1)+b0r+ c0 = 0 (32.14)

and this is called the indicial equation. (Note it is the indicial equation for the Euler equa-tion which comes from deleting all the nonconstant terms in the power series for p(x) andq(x).) Also the following equation must hold for n = 1, · · · .

p(n+ r)an =−n−1

∑k=0

ak (k+ r)bn−k−n−1

∑k=0

cn−kak ≡ fn (ai,bi,ci) (32.15)

These equations are all obtained by setting the coefficient of xn+r equal to 0.There are various cases depending on the nature of the solutions to this indicial equa-

tion. I will always assume the zeros are real, but will consider the case when the zeros aredistinct and do not differ by an integer and the case when the zeros differ by a non negativeinteger.