Math 257/316 Assignment 3
Due Friday Jan 30 in class
1. The steady-state temperature distribution y(x) along a wire 0 ≤ x ≤ 1, cooled by its
surroundings, and held fixed at zero temperature at its left endpoint, solves:
d
p(x)y 0 − y = 0,
y(0) = 0,
dx
where p(x) ≥ 0 represents its (spatially dependent) thermal conductivity. If the
conductivity degenerates at the left endpoint as p(x) = xs for some 0 ≤ s < 1, how
does the temperature behave near that point? (I.e. just give the form of the the first
term in a series solution about x0 = 0 (for x > 0).) Does the problem have a solution
if s = 1? s > 1?
2. Find (the first three non-zero terms of) a series solution (about x0 = 0) of this initial
value problem for a spherical Bessel equation:
x2
d2 y
dy
5
+ 2x
+ (x2 − )y = 0,
dx2
dx
16
y(0) = 0,
lim x3/4 y 0 (x) = 1.
x→0+
3. In the notes, one solution of the Bessel equation of order zero
x2 y 00 + xy 0 + x2 y = 0
is found to be
J0 (x) = 1 −
∞
X
1 4
1
(−1)m 2m
1 2
6
x
+
x
−
x
+
·
·
·
=
x .
22
22 42
22 42 62
22m (m!)2
m=0
Find a second (linearly independent) solution of the form
y2 (x) = J0 (x) ln(x) + ỹ(x)
as follows:
(a) show that for y2 to solve the Bessel equation, ỹ must solve the ode
x2 ỹ 00 + xỹ 0 + x2 ỹ + 2xJ00 (x) = 0;
(b) find the first three non-zero terms of a series solution ỹ(x) =
ode.
1
P∞
n=1 bn x
n
of this