M401 Spring 2010, Assignment 4 Solutions
1. [10 pts] In Problem 5 in Assignment 3, we used Taylors Theorem to nd the rst 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 a r
M401 Spring 2010, Assignment 3 Solutions
1. [10 pts] Exercise 1.5 on p. 15 of Simmonds and Mann Jr.
Solution. We rst 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 t
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 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 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
6
nx
3
(1)n 1 e 2 9 t cos
.
u(x, t) = +
22
2 n=1 n
3
Write down the two-te
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 dierence between this problem and the quarter-plane prob
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),
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 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 remo
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