59
AND FOURIER SERIES
be given oundarx value problem or else show that it has no
9 + 221 = 0, MO) = 1, W) = 0
$6." 1".3 If; 59 4/3531 (+):(frl'ncfw_a*l+fg 6:35CQ1
rf v/
r 3:!3
5C: \ _ ".,_L
jIU 26;! 3(0). c2 Q :1. 5c; =ff2.
z r 2.630 Va rr r Pu;
r e 4" U
S*Q^do
Math 2O4:
",o
Midterm Exam
#I
Fall 2008
.
Write your name and Student ID number in the sp&re provided below and sign.
Name and Last Name:
ID Number:
Signature:
.
Specify the section you have registered for by maxking one of the boxes provided below
,f "!t zoo
F^.cfw_ Cxarrn
h"J.L ro'cfw_
Problem 1. Find the \,alue of o such that the following equation is exact artd solve the
equation using this ralue of
a.
(xy2
lol
Nl?,Jl"
,t
(15 points)
+ a r2g)d,x + (x +
,ar<,n'i r.J
y)r2 dy
:
g
Altq,sl" 1l+g;r2
haLl 2r\. F\t %"9, F.n"\
Prcbbm r. Find
the
6xqvy,
ghdal &luti@ of th diflelenrial eluaiion:
t!'  (1+t)v' +v:
te!,
t>
o,
6m thaf h(t) = e'is a $lotion of the qstn\: 4/ \L+41/ +! = o lot
1> 0. (20 poi.is)
Y= e&, f . e (u+t^t
s"L.hbcfw_. 4 +r, h.^.n._,
MATH204: Midterm Exam#2. 5 May 2009
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Signature:
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Section1:
TuThu: 1112.15.
O
Section2:
TuThu: 1415.15
Duration of the Exam: 90 Minutes. (18.3020.00)
You must show all your work. You can use any statement which has
been proven
MATH204: Midterm Exam#123 March 2009
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Signature:
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Section1:
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Section2:
TuThu: 1415.15
Duration of the Exam: 90 Minutes. (18.3020.00)
You must show all your work. You can use any
MATH204: Final Exam: June 03, 2009
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Section1:
TuThu: 1112.15.
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Section2:
TuThu: 1415.15.
Duration of the Exam: 2h.30 Minutes. (1517.30)
You must show all your work. You can use any statement which has
been proven
, F"!( zo "8

2q
fr)
^14
nf4/.4'r. x"^, #z
Problem 1. Find the general solution of the following equations.
l.a)
4r2ytt
(,7 y=;n
=t

't'l = h(n1,)xcfw_z
,
,t rl'* s1 =pn4^,)'2
,
(4n 
So
(15 poinrs)
yl.nxil
=t
4x2\"
:g
4rgt + 5y
s
o
x
,<:7
ir\
8
7.4
3. Equation (14) states that the Wronskian satisfies the first order linear ODE
dW
= (p11 + p22 + + pnn )W.
dt
The general solution of this is given by Equation (15):
W (t) = C e
R
(p11 +p22 +pnn )dt
,
in which C is an arbitrary constant. Let X1 and X
7.7
1.(a) The eigenvalues and eigenvectors were found in Problem 1, Section 7.5.
1
2
(1)
(2)
r1 = 1, =
; r2 = 2, =
.
2
1
The general solution is
x = c1
e
2e
t
t
Hence a fundamental matrix is given by
e
(t) =
2e
+ c2
2 e2t
.
e2t
2 e2t
.
e2t
t
t
(b) We
HW # 6
Homework 7 Solutions
Section 5.1
130
Chapter 5. Series Solutions of Second Order Linear Equations
The series converges absolutely for x < 1 . Termbyterm dierentiation results in
1
X
y0 =
n3 x n
1
= 1 + 8x + 27x2 + 64 x3 + . . .
n=1
y 00 =
1
X
n
3.1
2. Let y = e". Substitution of the assumed solution results in the characteristic
equation 7'2 + 57' + 6 = 0. The roots of the equation are 7' = 3, 2. Hence the
general solution is y = ole2t + 0263t.
4. Substitution of the assumed solution 3; = 6" res
By definition of Convolution,
By Fubinis theorem we can switch the integration,
Look at the inner integral, by translation invariant
So we have shown that
242
Chapter 6. The Laplace Transform
Applying Theorem 6.6.1,
L
1
s
=
(s + 1)(s2 + 9)
10. We first no
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By definition of Convolution,
By Fubinis theorem we can switch the integration,
Look at the inner integral, by translation invariant
2. e2". = 6261. = 62(COS 1 2' sin 1).
3. e3"r = cos 37r+i sin 371' = 1.
10. The characteristic equation is r2 + 47' + 5 = 0 , with roots 1' = 2 :1: 2'. Hence
the general solution is y = Cl 62t cos t + C2 6 sin t.
20. The characteristic equation is 7'2 + 1
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x = c1
2
2t 1/2
+ c2
1
t
1 1/2
1 2
+
t2 0
t 0
4 ln t
6. The eigenvalues of the coefficient matrix
#HW5
1. The dierential equation is in standard form. Its coefficients, as well as the
function 4.1
g(t) = t , are continuous
everywhere.
HenceOrder
solutions
are Equations
valid on the
Chapter
4. Higher
Linear
Section
entire real line.
Writing the
equatio