Some Review Problems
Here are a few problems that are practice using the techniques we have developed in class. These are
by no means all of the types of problems that will appear on the exam but they are good practice for
the kinds of problems from chapt
Problem 1 Lets consider the following problem:
= c2 uxx a2 u
utt
u(0, t)
= ux (l, t) = 0
u(x, 0)
= f ( x)
ut (x, 0)
= g (x)
Here a and c are some real constants, l is a positive constant, and f and g are arbitrary functions.
To do this problem lets do the
Section 1.3 Solutions
1.
(a) Consider the equation
yuxx + uy = 0
Make the gues that u(x, y ) = X (x)Y (y ) for some unknown functions X and Y . Plugging this
guess into the equation yeilds:
yX (x)Y (y ) + X (x)Y (y ) = 0
Separating the terms depending on
Section 1.2 Solutions
1. To solve this problem we use the fact that L is a linear operator. We assume that L(u1 ) = f and that
L(u2 ) = f also. Now L(c1 u1 + c2 u2 ) = c1 L(u1 ) + c2 L(u2 ) = c1 f + c2 f = (c1 + c2 )f due to the fact that
L is linear. If
Section 5.5
2. To solve this problem we rst need to write down the actual BVP we wish to solve. Here we are dealing with the Heat equation in 3 spacial dimensions (cylindrical coordinates). We are also to assume
that the heat constant is equal to 1. Thus
Section 3.1
1. To prove the fact stated in the question we use the properties of the norm in terms of the corresponding
inner product. Namely:
a 2 = a, a
a+b
2
= a + b, a + b = a, a + 2Re a, b + b, b
ab
2
= a b, a b = a, a 2Re a, b + b, b
Here we have use
Section 2.1
4. Verify the formulas for entries 4 and 16 in Table 1 in Section 2.1 by hand using the denition of the
Fourier coefcients introduced in this section.
Recall that, for a 2 -periodic, integrable function f we have
an
bn
1
1
=
=
f () cos n d,
f
Section 1.1 Solutions
1. To show that u(x, t) = t1/2 exp(x2 /4kt) satises the heat equation ut = kuxx , we rst compute the
following partial derivatives of u(x, t).
ut
ux
uxx
2
2
x2 1/2 x2 /4kt
1
x2 5/2 x2 /4kt
1
t
e
= t3/2 ex /4kt +
t
e
= t3/2 ex /4kt +
M ATH 461 E XAM 2
May 28, 2008
N AME :
1. Please turn off all cell phones and pagers and remove all headphones.
2. There are 5 questions. You are given several pieces of paper to write your solutions on. Indicate clearly
which problem you are solving. If
M ATH 461 E XAM 1
April 30, 2008
N AME :
1. Please turn off all cell phones and pagers and remove all headphones.
2. There are 6 questions. You are given several pieces of paper to write your solutions on. Indicate clearly
which problem you are solving. I