PHYS 201 Mathematical Physics, Fall 2015, Homework 7
Due date: Thursday, December 3rd, 2015
1. Find the Frobenius series expansion about x = 0 of the solutions of the equation
2xy 00 − y 0 + x2 y = 0
2. This exercise follows Arfken 10.5.3(a), 10.5.4 to 10.5.7.
i. Show that the Green’s function for the operator
dy(x)
d
x
Ly(x) =
dx
dx
is
(
− ln t,
G(x, t) =
− ln x,
0≤x<t
t<x≤1
(1)
Due to the pathology at the boundary point, the solution to the inhomogeneous
equation involving L in terms of the Green’s function has an extra boundary term.
ii. Show Green’s theorem in one dimension for a Sturm-Liouville type operator:
Z b
dv(t)
d
du(t)
d
p(t)
− v(t)
p(t)
dt
u(t)
dt
dt
dt
dt
a
b
du(t) dv(t)
− v(t)p(t)
= u(t)p(t)
dt
dt a
iii. Using Green’s theorem in the form above, let
d
dy(t)
v(t) = y(t) and
p(t)
= −f (t)
dt
dt
d
∂G(x, t)
u(t) = G(x, t) and
p(t)
= −δ(x − t)
dt
∂t
and show that it yields
t=b
Z b
dy(t)
∂
y(x) =
G(x, t)f (t)dt + G(x, t)p(t)
− y(t)p(t) G(x, t) dt
∂t
a
t=a
iv. For p(t) = t, y(t) = −t and G(x, t) as in eq. 1, verify that the integrated part does
not vanish.
1
3. Solve the following initial value problem for t > 0 in terms of Green’s functions:
d2 y
dy
+ α + βy = f (t),
2
dt
dt
y(0) = A,
4. Compute the inverse Laplace transform of
f (s) =
k2
s(s2 + k 2 )
i. By expanding in partial fractions, and
ii. From the calculus of residues.
2
dy
(0) = B
dt