Physics 116C
Solutions to Homework Set #1
Fall 2012
1. Boas, problem p.564, 12.1-8
Solve the following dierential equations by series and by another elementary method
and check that the results agree:
(x2 + 2x)y 2(x + 1)y + 2y = 0
Series solution: substi
Physics 116C
Solutions to Homework Set #9
Fall 2012
1. Boas, p. 755, problem 15.6-13
Given a joint distribution function f (x, y ), show that E (x + y ) = E (x) + E (y ) and
Var(x + y ) = Var(x) + Var(y ) + 2 Cov(x, y ).
We only need to remember the denit
Physics 116C
Solutions to Homework Set #4
Fall 2012
1. Boas, p. 612, problem 12.22-5
Solve the Hermite dierential equation by power series
y 2xy + 2py = 0 .
n=0
We insert the power series y =
(1)
an xn into eq. (1), which yields
n 2
n(n 1)an x
2
n=2
n
an
Physics 116C
Solutions to Homework Set #3
Fall 2012
1. Boas, problem p.594, 12.16-6
Find the solution to the following dierential equation in terms of Bessel functions:
4xy + y = 0 .
(1)
Following eq. (16.1) on p. 593 of Boas, we rewrite eq. (1) in the fo
Physics 116C
Solutions to Homework Set #6
Fall 2012
1. Evaluate the integral
cos
d .
|r r |
(1)
The addition theorem for spherical harmonics is
4
P (cos ) =
2 + 1
Ym (, )Ym ( , ) ,
m=
where is the angle between the vectors characterized by the angles (,
Physics 116C
Final Exam Solutions
Fall 2012
1. The associated Laguerre polynomials are denoted by Lk (x). Evaluate Lk (0) in terms
n
n
of n and k .
According to eq. (22.25) on p. 610 of Boas, the associated Laguerre polynomials are
dened as
dk
Lk (x) = (1
Physics 116C
Solutions to Homework Set #2
Fall 2012
1. Boas, problem p.578, 12.7-5
Show that
1
1
P (x)dx = 0, for > 0.
This is immediate once we remember that P0 (x) = 1. It then follows that
1
1
1
P (x)dx =
1
1
P (x)dx 1 =
P (x)P0 (x)dx = 0 , for = 0 ,
1
Physics 116C
Midterm Exam Solutions
Fall 2012
1. Consider the dierential equation
2x(x + 1)y + 3(x + 1)y y = 0 .
(1)
(a) Using the Frobenius method, obtain the two linearly independent solutions of eq. (1).
If a solution takes the form of a generalized po
Physics 116C
Solutions to Homework Set #5
Fall 2012
1. Boas, p. 632, problem 13.3-4
At t = 0, two at slabs each 5 cm thick, one at 0o and one at 20o , are stacked together, and
then the surfaces are kept at 0o . Find the temperature as a function of x and
Physics 116C
Solutions to Homework Set #8
Fall 2012
1. Boas, p. 734, problem 15.3-6
A card is selected from a shued deck. What is the probability that it is either a king or
a club? That it is both a king and a club?
The probability of having either a kin
Physics 116C
Solutions to Homework Set #7
Fall 2012
1. Boas, p. 658, problem 13.8-4
Do the two-dimensional analogue of the problem in Boas, p.655, Example 1. That is, solve
the Poisson equation in two dimensions, with a point charge outside a grounded cir