PHZ 3113 Fall 2011 – Homework 12
Due at the start of class on Monday, December 5. No credit will be available for
homework submitted after the deadline.
Answer all three questions. Please write neatly and include your name on the front page of
your answers. You must also clearly identify all your collaborators on this assignment. To
gain maximum credit you should explain your reasoning and show all working.
1. Consider a solution of the differential equation
in the series form y =
P∞
n=0
d2 y
dy
+y =0
+x
2
dx
dx
an xn+s with a0 6= 0.
(a) Calculate and solve the indicial equation to find two allowed values of s, the lowest
power of x present in the solution.
(b) For each allowed value of s, find a recurrence relation for the coefficients an and
hence find an /a0 .
(c) For each allowed value of s, find the interval of convergence of the corresponding
series solution.
2. Consider a solution of the differential equation
x(x − 1)
in the series form y =
P∞
n=0
d2 y
dy
+y =0
+ 3x
2
dx
dx
an xn+s with a0 6= 0.
(a) Calculate and solve the indicial equation to find two allowed values of s, the lowest
power of x present in the solution.
(b) For each allowed value of s, find a recurrence relation for the coefficients an .
(c) Show that one of the two allowed values of s leads to a series solution with welldefined an /a0 that converges within some interval of x.
(d) Show that the other allowed value of s does not lead to a valid series solution
because a1 /a0 is infinite.
For your information: Fuch’s theorem states that a homogeneous second-order differential equation of the form
d2 y
dy
+
p(x)
+ q(x)y = 0
dx2
dx
P∞ (k)
n+s
if and
has two solutions in terms of Frobenius series Sk (x) =
n=0 an (x − x0 )
only if (x − x0 )p(x) and (x − x0 )2 q(x) both have convergent power-series expansions
about x = x0 . If p(x) and q(x) both have locally convergent power-series expansions
about x = x0 , then x0 is a regular point and the linearly independent solutions of
the differential equation are of the form y = S1 (x) and y = S2 (x). Otherwise, x0
is a regular singular point and the linearly independent solutions are y = S1 (x) and
y = S1 (x) ln(x − x0 ) + S2 (x). The second case is illustrated by the question above.
3. Consider a thin, rectangular metallic plate extending over (x, y) coordinates 0 < x <
W and 0 < y < L. Along the y = 0 edge, the plate’s temperature T (x, y) varies
linearly from T (0, 0) = T0 to T (W/2, 0) = T1 , and linearly from T (W/2, 0) = T1 to
T (W, 0) = T0 . The other three edges of the plate are held at temperature T0 . By
solving Laplace’s equation
∂ 2T
∂2T
+
= 0,
∂x2
∂y 2
find the full temperature distribution T (x, y) within the plate.