Extra problem 1
Match the direction elds below with the rollowing equations. Each numbered equation corresponds to a dierent lettered graph
1) y = 2
y
5) y =
x
2) y = x
6) y =
x
y
3) y = y x
y2
7) y
Mat 244 Assignment 2
Due July 10.
1. [1 point] Check that
y(t) =
1
t
f (s) sin(t s)ds
0
is a solution of the initial value problem
y (t) + 2 y(t) = f (t),
> 0,
y(0) = 0, y (0) = 0.
2. [2 points] Solv
In this note we present a fast method for solving non-homogeneous higher
order dierential equations with constant coecients. While such methods
as undetermined coecients and variation of parameters ar
Mat 244 Assignment 1
June 6, 2013
Due June 12.
The assignment probably looks long, but the opposite is true. It looks long
just because there is a lot of explanation of how to write solutions to the q
Existence and Uniqueness Theorem
Elements of the Real Analysis
Denition 1. Let
(i) C 0 ([a, b]) := cfw_f : [a, b] R | f is continuous be a space of continuous functions on [a, b] with a norm f max[a,b
Linear ODEs with constant coecients
Let
n
L(y) = y
(n)
pj y (nj) ,
+
j=1
n
def
Q(z) = z n +
pj z nj .
j=1
Claim 1.
m
L
x
erx
m!
= erx
0smin(m,n)
Lemma 2.
j
(j)
(f g)
(s)
ms
Q (r)
x
(m s)!
s!
j (s)
JVIat 244, , Final exam Page 7 of 17
6. (10 points) Variation of parameters
(a) [4 points] Let y1,y2 be two solutions of
y + W)?! + (my = 0,
and let W be their Wronskian. Derive a differential equ
Extra problem 2
Determine which of the numbered graphs represents the family of solutions
to each of the following equations
a) y = y 2 1
y
d) y = 2
x 1
b) y = 2x + y
sin(3x)
e) y =
1 x2
c) y = sin(x)
Mat 244 Assignment 3
August 6, 2013
Due August 7.
1. [3 points] Find the curve y going from (0, 0) to (L, h) (i.e. y(0) = 0
and y(L) = h) that when rotated around the x-axis gives a surface that has
l