MACT 2141
Fall 2019
Final Exam
Dec 14, 2019
Name:
Circle your section:
UID:
1
2
3
4
• Time: 2 hours.
• Read each question carefully.
• Show your work to receive full credit.
• A two-sided formula sheet prepared by the student is allowed.
• Only non-graphing and non-programmable calculators are allowed.
Problem Score Points
1
20
2
30
3
20
4
20
5
15
Total
105
MACT 2141
Fall 2019
Laplace Transform Formulas
Definition of the Laplace Transform:
Z ∞
F (s) = L[f (t)] =
e−st f (t) dt
0
Transforms of Elementary Functions:
L [tn ] =
n!
sn+1
s
L [cos(at)] = 2
s + a2
s
L [cosh(at)] = 2
s − a2
1
s−a
a
, L [sin(at)] = 2
s + a2
a
, L [sinh(at)] = 2
s − a2
, L eat =
Properties of the Transform:
L [c1 f1 (t) + c2 f2 (t)] = c1 F1 (s) + c2 F2 (s)
L eat f (t) = F (s − a)
L [f (t − a)U (t − a)] = e−as F (s)
L [f 0 (t)] = sF (s) − f (0)
L [f 00 (t)] = s2 F (s) − sf (0) − f 0 (0)
n
n
n d
L [t f (t)] = (−1)
F (s)
n
ds
Z t
1
L
f (τ )dτ = F (s)
s
0
L [f (t) ∗ g(t)] = F (s)G(s)
The Unit Step Function: U (t − a) =
(
0 if t < a
1 if t ≥ a
Z t
The Convolution: f (t) ∗ g(t) =
f (τ )g(t − τ ) dτ
0
MACT 2141
Fall 2019
Problem 1. (10 pts each) Consider the differential equation:
dy
+ 2y = f (t)
dt
a. Letting f (t) = t, find the solution y(t) of the above DE that satisfies the initial
condition y(0) = α. (You need to get y(t) in terms of α.)
b. Show that if y1 (t) and y2 (t) are solutions of the above DE for any f (t), then:
lim [y1 (t) − y2 (t)] = 0.
t→∞
MACT 2141
Fall 2019
Problem 2a. (10 pts) Find the general solution of the differential equation:
y 00 + 4y 0 + 5y = 5e−2t
MACT 2141
Fall 2019
Problem 2b. (10 pts) Find the general solution of the differential equation:
e−2t
y + 4y + 4y =
t
00
0
MACT 2141
Problem 2c. (10 pts) Find the solution of the initial value problem:
(
0
t<1
y 00 + 5y 0 + 6y =
,
y(0) = 0, y 0 (0) = 0.
−2t
e
t≥1
Fall 2019
MACT 2141
Fall 2019
Problem 3a. (5 pts each) Find the following Laplace and inverse Laplace transforms:
i. L cos2 (5t) =
Z t
ii. L
xe
x−t
dx =
0
iii. L−1
se−s
=
s2 + 2s + 5
MACT 2141
Fall 2019
Problem 3b. (5 pts) Use Laplace transform to evaluate the integral:
Z ∞
e−2x x sin x dx.
0
MACT 2141
Fall 2019
Problem 4a. (10 pts) Solve the following system of differential equations:
dx
+ y = t,
dt
dy
4x +
= 0,
dt
with the initial conditions: x(0) = 1, y(0) = 2.
MACT 2141
Fall 2019
Problem 4b. (10 pts) Find power series solution around x = 0 of the initial value
problem:
y 00 − xy = 0,
y(0) = 1, y 0 (0) = −1
MACT 2141
Fall 2019
Problem 5. (5 pts each) True or False (Circle one and state your reason):
a. The following initial value problem has a unique solution:
y 000 + y sin(t) = 5,
y(3) = 5,
y 0 (3) = −2,
Reason:
y 00 (3) = 4
True
False
True
False
True
False
b. The Laplace transform of the function cos(t2 ) exists.
Reason:
c. The Laplace transform of the function cosh(t2 ) exists.
Reason:
MACT 2141
Draft:
Fall 2019