M401 Spring 2010, Assignment 1, due Thursday January 28
1. [10 pts] In many introductory textbooks on ODE, air resistance is modeled with a linear
term (because this gives rise to equations that are easy to solve). In this case the equation
for a falling
M401 Spring 2010, Assignment 4 Solutions
1. [10 pts] In Problem 5 in Assignment 3, we used Taylors Theorem to find the first two
terms in a perturbation expansion of the solution of the ODE
y + k 2 y = y 2 ,
y(0) = 1
y (0) = 0.
1a. Write this equation as
Math 401 (Sec. 502) Homework 4 (due February 19)
Spring, 2009
Q1. Consider the following IVP (called damped linear oscillator) y + 2y + y = 0; (a) Compute the exact solution. (b) Use two-scale perturbation theory to find a leading-order approxima
Math 401 (Sec. 502) Homework 5 (due February 26)
Spring, 2009
Use boundary-layer theory to obtain a one-term uniformly valid expansion for the solutions of Q1. y + y + e-t = 0; Q2. y + ty + y = 0; Q3. y + (t + 1)y + y = 2t; y(0) = 0, y(1) = 1
M401 Spring 2010, Assignment 5 Solutions
1. [10 pts] We saw in Problem 5 from Assignment 3 that for
y + k 2 y = y 2
y(0) = 1
y (0) = 0,
the function y1 (t) in the perturbation expansion y(t) = y0 (t) + y1 (t) + . . . does not have
a secular term to remove
M401 Spring 2010, Assignment 6 Solutions
Note on Problems 13. In approximating solutions of equations of the form
y + k 2 y = f (y, y )
with the two-scale method its often convenient to use polar form, especially when f (y, y ) is
not odd in y . In Proble
M401 Spring 2010, Assignment 1, due Thursday January 28
1. [10 pts] In many introductory textbooks on ODE, air resistance is modeled with a linear
term (because this gives rise to equations that are easy to solve). In this case the equation
for a falling
M401 Spring 2010, Assignment 9, due Thursday April 15
1a. [6 pts] Solve the quarter-plane problem
utt = c2 uxx ; (x, t) (0, ) (0, )
ux (0, t) = 0; t 0
u(x, 0) = f (x); x 0
ut (x, 0) = g(x); x 0.
Notice that the difference between this problem and the quar
M401 Spring 2010, Assignment 3 Solutions
1. [10 pts] Exercise 1.5 on p. 15 of Simmonds and Mann Jr.
Solution. We first observe that for = 0, we have x(0)2 + 1 = 0 x(0) = 1, and since
these roots are not repeated the Implicit Function Theorem will hold for
M401 Spring 2010, Assignment 4, due Thursday February 18
Note. Save this assignment page. You will be allowed to bring it to both the midterm and
the final. (No, you cant bring solutions.)
1. [10 pts] In Problem 5 in Assignment 3, we used Taylors Theorem
M401 Spring 2010, Assignment 5, due Thursday Feb. 25
1. [10 pts] We saw in Problem 5 from Assignment 3 that for
y + k 2 y = y 2
y(0) = 1
y (0) = 0,
the function y1 (t) in the perturbation expansion y(t) = y0 (t) + y1 (t) + . . . does not have
a secular te
M401 Spring 2010, Assignment 11, due Thursday April 29
1a. [5 pts] Compute the Fourier cosine series for f (x) = sin x on [0, ].
1b. [3 pts] Find an upper bound on the error obtained if the first N terms of the Fourier
cosine series from Part (a) are used
M401 Spring 2010, Assignment 11 Solutions
1a. [5 pts] Compute the Fourier cosine series for f (x) = sin x on [0, ].
Solution. Since L = , the series will have the form
f (x) = a0 +
X
an cos nx,
n=1
where
1
1
2
1
a0 =
sin xdx = ( cos x) = ( cos + 1) =
0
M401 Spring 2010, Assignment 6, due Thursday March 4
Note on Problems 13. In approximating solutions of equations of the form
y + k 2 y = f (y, y )
with the two-scale method its often convenient to use polar form, especially when f (y, y ) is
not odd in y
M401 Spring 2010, Practice Problems for Final Exam
The final exam for this class will be on Wednesday, May 12, 8:00-10:00 a.m., in Blocker 161
(the regular classroom). It will cover the method of characteristics (including data given on
the x-axis, on a q
M401 Spring 2010, Assignment 3, due Thursday February 11
1. [10 pts] Exercise 1.5 on p. 15 of Simmonds and Mann Jr.
2. [10 pts] Find the first two terms in the expansions of each of the four roots of
x4 + x3 x2 + 2x 1 = 0,
for 0.
3. [10 pts] Solve the ODE
M401 Spring 2010, Assignment 9 Solutions
1a. [6 pts] Solve the quarter-plane problem
utt = c2 uxx ; (x, t) (0, ) (0, )
ux (0, t) = 0; t 0
u(x, 0) = f (x); x 0
ut (x, 0) = g(x); x 0.
Notice that the difference between this problem and the quarter-plane pro
M401 Spring 2010, Assignment 8, due Thursday April 8
1. [10 pts] Constanda Exercise 12.3, Parts (i) and (ii).
2. [10 pts] Constanda Exercise 12.4, Parts (i) and (ii).
3. [10 pts] Constanda Exercise 12.5, Parts (i) and (ii).
4. [10 pts] Constanda Exercise
M401 Spring 2010, Assignment 7 Solutions
1a. [5 pts] Non-dimensionalize the wave equation
utt = c2 uxx ; (x, t) (0, L) (0, )
u(0, t) = U1 ; u(L, t) = U2 ; t > 0
u(x, 0) = f (x); x [0, L]
ut (x, 0) = g(x), x [0, L].
Solution. We set
=
t
,
A
x
,
B
=
v(, ) =
M401 Spring 2010, Assignment 10, due Thursday April 22
1a. [5 pts] In class we solved
1
ut = uxx
2
ux (0, t) = 0; ux (3, t) = 0;
u(x, 0) = x, x [0, 3].
t0
and we found the solution to be
1 n2 2
nx
3 X 6
n
(1)
1
e 2 9 t cos
.
u(x, t) = +
2
2
2 n=1 n
3
W
M401 Spring 2010, Assignment 8 Solutions
1. [10 pts] Constanda Exercise 12.3, Parts (i) and (ii).
Solution to Part (i). In order to be consistent with our calculations from class Ill write
the equation as
1
1
ux + uy = u.
2
2
Now, set U(x) = u(x, y(x), so
M401 Spring 2010, Assignment 2, due Thursday February 4
1. [10 pts] In Assignment 1 we used Taylors Theorem to solve Exercise 1.4 on p. 12 of
Simmonds and Mann Jr. Solve the same problem using the expansion method.
2. [10 pts] Use the expansion method to
M401 Spring 2010, Assignment 10 Solutions
1a. [5 pts] In class we solved
1
ut = uxx
2
ux (0, t) = 0; ux (3, t) = 0;
u(x, 0) = x, x [0, 3].
t0
and we found the solution to be
1 n2 2
nx
3 X 6
n
(1)
1
e 2 9 t cos
.
u(x, t) = +
2
2
2 n=1 n
3
Write down the
M401 Spring 2010, Assignment 2 Solutions
1. [10 pts] In Assignment 1 we used Taylors Theorem to solve Exercise 1.4 on p. 12 of
Simmonds and Mann Jr. Solve the same problem using the expansion method.
Solution. We have already checked that the Implicit Fun
M401 Practice Problems for Midterm Exam
The midterm exam will be Thursday, March 11, 7:00-9:00 p.m. in Blocker 161 (our regular
classroom). Calculators will be allowed on the exam, and you can use a copy of Assignment
4. The exam will cover dimensional an