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 =
x1
4) y = y
8) y = xy 2 + x2
(a)
(b)
(c)
(d)
(e)
(f)
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] Solve the initial value problem
y + 4y + 4y =
e2t
,
t2
y(1)
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 are valid techniques
to solve such equations, they are of
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 questions. You are being trained to write readable solut
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] |f (x)| and a distance
dist(f, g) f g = max[a,b] |f (
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) (js)
f g
.
s
=
s=o
Proof. Proof by induction on j start
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 equation that W satises.
(b) [6 points] Solve the initial
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) sin(y)
One of the given graphs does not match any equa
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
least total resistance moving in a ow of gas. Take as gr