University of Georgia
Department of Mathematics
MATH 2700 – Test #3 - Sample
Elementary Differential Equation
Instructor: Dr. Lin Mu
November 14, 2022
Name:
Student Number:
This exam contains 7 pages (including this cover page) and 5 questions. Total of points is 100.
Good luck and Happy solving differential equation problems!
1
MATH 2700
Test #3 - Sample
November 14, 2022
Distribution of Marks
Question:
1
2
3
4
5
Total
Points:
20
20
25
16
19
100
Bonus Points:
0
10
0
0
0
10
Score:
1. For the given equation,
y 00 + 5y 0 + 6y = e−2t
(a) (10 points) Find the general solution.
(b) (5 points) Describe the long time behavior of the solution for t → ∞.
(c) (5 points) Do you have different predictions for part b) for different initial conditions.
Page 2 of 7
MATH 2700
Test #3 - Sample
November 14, 2022
2. For the given equation
d2 y
+ 5y = 3 cos(2t),
dt2
(a) (10 points) Find the solution.
(b) (5 points) Find the phase angle for the particular solution in part a).
(c) (5 points) Describe the long time behavior of the solution of part a).
(d) (10 points (bonus)) Determine the frequency. Do you expect resonance in this undamped
forcing system?
Page 3 of 7
MATH 2700
Test #3 - Sample
November 14, 2022
3. Suppose a spring with spring constant 1 N/m is horizontal and has one end attached to a wall
and the other end attached to a 4 kg mass. Suppose that the friction of the mass with the floor
(i.e., the damping constant) is 4 N · s/m.
(a) (10 points) Set up a differential equation that describes this system. Let y denote the
displacement (in meters), of the mass from its equilibrium position. Assume the positive displacement means the mass is farther from the wall than when the system is at
equilibrium.
(b) (10 points) Find the general solution to your differential equation from the previous part.
(c) (5 points) Is this system under damped, over damped, or critically damped?
Page 4 of 7
MATH 2700
Test #3 - Sample
4. Find the Laplace transform of (Calculate the transform by definition.)
(a) (8 points) f (t) = 1.
(b) (8 points) f (t) = t.
Page 5 of 7
November 14, 2022
MATH 2700
Test #3 - Sample
5. Find the inverse Laplace transform of the given functions,
1
.
(a) (9 points)
s−3
4
.
(b) (10 points)
s(s + 3)
Page 6 of 7
November 14, 2022
MATH 2700
Test #3 - Sample
This page is left blank for extra scratch.
Page 7 of 7
November 14, 2022