Differential Equations - Practice Problems
Section 5.5. Series Solutions Near a Regular Singular Point
Problem 5-1. Consider the following equation
2x2
d2 y
dy
+ x (1 + x)y = 0.
2
dx
dx
(a) Show that x = 0 is not a regular point, but a regular singular po
NAME (please print):
FINAL EXAM (SAMPLE 1)
1. Which of these PDEs is the heat equation, the wave equation, or the Laplaces
equation? Write the name of the equation next to it.
utt = 2uxx ,
ut = 3uxx + 3,
2ut = 3uxx ,
2uxx + 2uyy = 0.
2. (a) How many i
Practice Quiz 3
1. State the principle of superposition for non-homogeneous problems L(u) = f ,
f 6= 0.
2. (a) Find the eigenvalues and eigenfunctions of the eigenvalue problem
d2
dx2
(0)
=
0,
(2)
(L)
=
0.
(3)
= ,
> 0,
(1)
(b) How many eigenvalues are t
Quiz 2 Sample
1. Write the balance law governing the rate of change of energy for a rod of
length L, with non-zero source term Q.
2. Determine the equilibrium temperature distribution for a one-dimensional
rod with constant thermal properties with the fol
Practice Quiz 4
1. True of false: Zero is an eigenvalue of the following eigenvalue problem:
d2
dx2
(0)
=
0,
(L)
=
0.
= ,
2. What is the minimum eigenvalue and the corresponding eigenfunction of the
following eigenvalue problem:
d2
= ,
dx2
d
(0) = 0,
dx
Practice Quiz 8
1. (a) What does the mean value property for the Laplaces equation state?
Prove the mean value property.
(b) What is does the Maximum Principle for the Laplaces equation state?
2. How do you find the general solution G(r) of the following
Quiz 1 Problems Sample
Quiz 1 will cover Sections 1.1, 1.2, and 1.3 from the text book (pages 1-14).
1. What is Newtons Law of Cooling? State it for both ends of the rod (i.e., at
x = 0 and x = L).
2. Write the Initial Boundary Value problem modeling heat
Practice Quiz 6
1. Write DAlambert formula for the solution of an initial-value problem for
the wave equation utt = c2 uxx , with the initial displacement given by f (x) and
initial velocity given by g(x).
2. What is the form of the initial-value problem
NAME (please print):
Practice Midterm Exam
1. Write the definition of the total heat energy of a rod of length L. Explain
each symbol in the formula you write.
2. Consider the problem:
ut = uxx + x ,
u(x, 0) = f (x), ux (0, t) = 0, ux (L, t) = 0.
For whic
Practice Quiz 5
1. True of false:
(a) Solutions of the wave equation utt = c2 uxx decay in time.
(b) Solutions of the heat equation ut = kuxx decay in time.
2. In the wave equation utt = c2 uxx describing vibrations of an elastic string,
what is the meani
Practice Quiz 7
1. What happens with the total energy E of a vibrating string if
and u(L, t) = 0?
u
x (0, t)
=0
(Show your work.)
2. Solve Laplaces equation inside a rectangle 0 x L, 0 y H, with the
following boundary conditions:
u
2y
(0, y) = 0, u(L, y)