18.03 Problem Set 4b Spring 2009
Due with Problem Set 4a in Room 2106 at 12:55 pm, Friday, March 6
Part A
20 points for PS4a (Part A) + PS4b (Part A)
0. (at due date)
Write the names of any person, web site, or materials you consulted.
Write “No C” (no consultation) on your paper if you consulted no outside materials/people.
3. (Lec 11, Fri Feb 27: Solving
y
00
+
ay
0
+
by
= 0
,
y
(
x
0
) =
y
0
,
y
0
(
x
0
) =
v
0
)
Read: EP 2.1 (Skip Theorems 2, 3; pay attention to the last three pages 109–111, constant
coeﬃcient equations); EP (p. 129–132; Complexvalued functions, Examples 2–5)
Work: 2C1abcde
4. (Lec 12, Mon Mar 2: Damping; complex roots )
Read: EP 2.3, 2.4 ; Work: 2C3, 5, 10ab
5. (Lec 13, Wed Mar 4: Main theorems about linear 2ndorder ODE’s)
Read: 2.1 Theorems 1, 2, 4; Work: 2A1a, 2, 3
Part B
48 points for the sum of 4a and 4b (+ 10 extra)
1. (on PS4a)
Godzilla = 12pts = 4 + 2 + 2 + 2 + 2. (
X. Extra credit
10 pts)
2. Lec 9 Linear approximation to nonlinear ODE.
(10 pts = 2 + 2 + 6)
a) Draw enough solution curves to the equation
y
0
= 3
y

y
2
= (3

y
)
y
to indicate all
behaviors, especially near the stable trajectory
y
= 3.
b) We will now explain why nearby solutions to
y
= 3 tend to 3 exponentially fast by
comparing to the linear case. Write nearby solutions as
y
= 3 +
u
where
u
is small. Derive
the equation for
u
,
u
0
=

3
u

u
2
c) The idea of linearization is that, since we expect
u
to be small as
x
→ ∞
, we omit the