ODE Practice Midterm Exam Solution
Problem 1 (20 pts)
Find the solutions of the given diﬀerential equations.
1. y − 4y + 4y = 0
2. (2x − y) + (2y − x)y (x) = 0
1. For y − 4y + 4y = 0, the charecteristic equation is r2 − 4r + 4 = 0. We have a repeated
root
ODE Midterm 1 Solution
Problem 1 (7 pts)
Find the general solutions of the given dierential equations.
1.
dy
dt
+ 2y = e2t . (4 pts)
The integrating factor saties f = 2f . We may choose f = e2t Mutiplying the integrating
factor, we have
(e2t y ) = 1
Integ
ODE Practice Midterm Exam
November 14 th, 2012
NAME (please print):
ID:
SCHOOL:
Write clearly and show all work.
This is a closed book test and no calculators are allowed.
Problem
Problem
Problem
Problem
Problem
Total
1
2
3
4
5
20 pts.
20 pts.
20 pts.
20
ODE Practice Midterm Exam
November 14 th, 2012
NAME (please print):
ID:
SCHOOL:
Write clearly and show all work.
This is a closed book test and no calculators are allowed.
Problem
Problem
Problem
Problem
Problem
Total
1
2
3
4
5
20 pts.
20 pts.
20 pts.
20
Problem 1 (20 pts)
Find the general solutions of the given dierential equations.
1.
dy
dt
+ y = et .
2. y y = e2t
1. The integrating factor satises the equation
f =f
Hence, we can choose it to be f = et . Mutiple the equation with the integrating factor,
ODE Practice Midterm Exam
November 14 th, 2012
NAME (please print):
ID:
SCHOOL:
Write clearly and show all work.
This is a closed book test and no calculators are allowed.
Problem
Problem
Problem
Problem
Problem
Total
1
2
3
4
5
20 pts.
20 pts.
20 pts.
20
Problem 1 (20 pts)
Find the general solutions of the given dierential equations.
1. y (6) + 4y = 0
2. y 2y + 3y 2y = et
1. First we solve Z 6 = 4. Write Z = rei . Then Z 6 = r6 ei6 . On the other hand,
4 = 4ei Hence,
r6 = 4
6 = + 2k
and we have r = 41/6 a
Problem 1 (6 pts)
Find the general solutions of the given dierential equations.
1. y (4) + 16y = 0. (3 pts)
2. y 5y + 6y = et sin t. (3 pts)
1. Characteristic equation is Z 4 = 16. Suppose Z = rei then
r4 ei4 = 16ei
Hence r = 2 and
4 = + 2k
The roots are
ODE Practice Midterm Exam
November 14 th, 2012
NAME (please print):
ID:
SCHOOL:
Write clearly and show all work.
This is a closed book test and no calculators are allowed.
Problem
Problem
Problem
Problem
Problem
Total
1
2
3
4
5
20 pts.
20 pts.
20 pts.
20