Math 308
Worksheet 4: Section 5.5
13 & 16 July 2012
Key Ideas. Start with a linear, second-order, homogeneous equation
y 00 + p(x) y 0 + q(x) y = 0 .
(I) If xo is an ordinary point, then we can find analytic solutions
∞
X
(1)
y =
an (x − xo )n .
n=0
(II) If xo is a regular singular point, then we can find a series solution
!
∞
X
an (x − xo )n .
(2)
y = (x − xo )r 1 +
n=1
The difference between the two solutions (1) and (2) is that the second solution may
not be analytic at xo : that is, r may be negative, rational (e.g. r = 3/2), or even
irrational (e.g. r = −π).
(III) If xo is irregular, then we do not expect to find series solutions.
Exercise (§5.4, #18, 20). Consider two linear, second-order, homogeneous equations
(3a)
(3b)
x2 (x − 1)2 y 00 + 2xy 0 + 4y = 0 ,
x2 (1 − x2 )y 00 + (2/x)y 0 + 4y = 0 .
Which of these equations can you find a series solution for at xo = 0? And at xo = 1?
Exercise (§5.5, #1). Find the first four nonzero terms of two series solutions to
2xy 00 + y 0 + xy = 0 at x = 0.
Homework. As part of your preparation for Quiz 2 (Wednesday, July 18) and the
Final Exam (Tuesday, August 07), I recommend the following:
(A) Section 5.5: Problems 2, 6, 8.
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Colleen Robles
robles@math.tamu.edu