1. (10 pts.)
f (t)
L {f (t)} = F (s)
a
a
s
e s
s(s + 1)2
sister
As
1
A S IF
n!
sn+1
tn
Bstet
s
a
s
cos kt
Let
k
s2 + k 2
eat f (t)
F (s
a)U (t
a)
a)
0
1
1
1
s
s2 + k 2
sin kt
5
0
2
clap
A
C
C
1
B
B
A
s
éᵗ tet
w
f
sister
L
L
CS
A
coeff of s
e as F (s)
42
Bs S 1
1
eat
f (t
Find the inverse Laplace transform of F (s) =
is
e
3
e
t
up
éÉ
oct
1
1
ce.net
uce n
1
2. (10 pts.)
Solve the following di↵erential equation
p
dy
=2+
2x + y + 3
dx
5
3. (10 pts.)
Consider the following system of di↵erential equations
x0 4y = 0
x + y0 = 1
x(0) = 2,
y(0) = 1.
Use the Laplace transform to solve for y(t) only.
f (t)
L {f (t)} = F (s)
a
a
s
sn+1
Y
4Y
70
s
a
s
cos kt
s2 + k 2
sin kt
eat f (t)
F (s
sF
SY
SI
I
k
s2 + k 2
a)
44
sy
3
f (0)
4
yet
yet
L
L
cost
Ys
Esinat
7
0
44
2
SI
I
y
0
sy yeol
I
1
eat
df
dt
SX
n!
tn
I L
Let
1
2
1
5
1
5
5
1
2
52 4
5h4
44
4
4. (10 pts.)
Find the inverse Laplace transform of F (s) =
f (t)
L {f (t)} = F (s)
a
a
s
tn
eat
cos kt
5 25 1 9
FCs
n!
sn+1
5
s
1
Is 15
1
a
s+2
s2 + 2s + 10
9
5 9
st112 9
s
s2 + k 2
sin kt
k
s2 + k 2
eat f (t)
F (s
a)
L
été
Fasig
e
cost
9
Fa éÉ
Jets inst
a
5. (10 pts.)
Suppose y1 (x) = e x and y2 (x) = xe x are solutions of the equation
y 00 + p(x)y 0 + q(x)y = 0. Find the general solution of the equation
y 00 + p(x)y 0 + q(x)y = e x ln x,
11
x > 0.
6. (10 pts.)
d4 y d3 y d2 y
+
+
=0
dx4 dx3 dx2
(a) Find the general solution of the following di↵erential equation
(b) Consider the following di↵erential equation
a2
d2 y
dy
+ a1
+ a0 y = 1 + x cos x
2
dx
dx
The general solution to the associated homogeneous equation is yc = c1 cos x + c2 sin x.
When using the method of undetermined coefficients, what form should one use for a particular solution yp ? DO NOT ACTUALLY FIND THE COEFFICIENTS. YOU ONLY HAVE TO DETERMINE
THE CORRECT FORM.
split
atefytas.de taoy
a
Yp
try
try
off
as
1
1
1 they
cos
12
A
yp
yp
ex
Ex
cosx
D
Tike term
cx
Dx
Yp Gp
Yp
13
in
cost
F
sinx
Ye
Ex
Fx
sinx
7. (10 pts.)
Use the method of undetermined coefficients to solve the following initial-value problem
y 00 + 5y 0 + 4y = 6e x ,
y(0) = 2, y 0 (0) = 6.
Note: The general solution to the associated homogeneous equation is yc = c1 e x + c2 e 4x .
15