MA 36600
MIDTERM I EXAM INSTRUCTIONS
NAME
1. NO CALCULATORS, BOOKS, OR PAPERS ARE ALLOWED. Use the back of the test
pages for scrap paper.
2. For All problems, show all your work and write (or mark) the answers clearly.
Your answers
1. (5 pts)
2. (5 pts)
3. (5 pts)
4. (5 pts)
5. (5 pts)
6. (5 pts)
7. (5 pts)
8. (5 pts)
9. (5 pts)
10. (6 pts)
11. (7 pts)
12. (7 pts)
13. (6 pts)
Total Points:
1
1. (5 points) Which of the following initial data will not guarantee a unique solution for the
problem
y 0 = (y + 2)2/3 + (t − 1)1/3 , y(t0 ) = y0 .
a. (t0 , y0 ) = (1, 2)
b. (t0 , y0 ) = (2, 0)
c. (t0 , y0 ) = (2, 2)
d. (t0 , y0 ) = (−1, 2)
e. (t0 , y0 ) = (−1, −2)
2. (5 points) Let y1 (x) = e2x , y2 (x) = xe2x , find the Wronskian of y1 (x) and y2 (x).
A. e4x
B. 0
C. 2xe4x
D. −2xe4x
E. 2e2x
2
3. (5 points) Using Euler method with h = 0.1 on
y 0 (t) = y(t)2 + t2 ,
The approximate value for y(1.2) is
y(1) = 1.
.
A. 2.5
B. 0
C. 1.465
D. −2.56
E. 3.778
4. (5 points) If y = xr is the solution to the equation
x2 y 00 + 2xy 0 − 6y = 0,
then r equals to
A. 1, or 2
B. 2, or − 2
C.
2, or − 3
D. −2, or − 3
E. 2, or − 4
3
x > 0,
5. (5 points) Let y = tr be a solution to
t2 y 00 − 2ty 0 + 2y = 0.
What are the values of r?
A. 1, or 2
B. 2, or − 2
C. 2, or − 3
D. −2, or − 3
E. 2, or − 4
6. (5 points) Find the solution of the following differential equation
u0 + 2u = 2xe−2x ,
A. u(x) = (x2 + 2)e2x
B. u(x) = (x2 + 2)e−2x
C. u(x) = (x2 + 2x + 2)e2x
D. u(x) = (−x2 + 2)e−2x
E. u(x) = x2 e2x + 2
4
u(0) = 2.
7. (5 points) Transform the following equation to a separable or linear equation and find
the general solution:
y
y2
y0 = − 2
x x
A. y =
x
ln |x|+cx
B. y =
x+c
ln |x|
C. y =
D. y =
E. y =
x
ln |x|+c
x
ln |x|
√
+c
2x − 1
8. (5 points) If y 00 − 2y 0 = 0;
y(1) = −1,
y 0 (1) = 2, then y(2) =
A. e2
D.
1 2
e − 12
2
2 2
e − 12
3
2 −2
e − 12
3
E.
e2 − 2
B.
C.
5
9. (7 points) Find the EXPLICIT solution of the following differential equation
(9x2 + y − 1)dx − (4y − x)dy = 0,
y(0) = 1
A. 3x3 + 2xy − x − 2y 2 = 0
p
B. y = − 41 (−x + x3 + 2xy − x)
√
C. y = 14 (−x + x2 − 24x3 + 8x + 16)
√
D. y = − 14 (−x + x2 − 24x3 + 8x + 16)
√
E. y = 2x − 1
10. (5 points) A culture is growing according to the Malthusian model (P 0 = rP ). It doubles
every two days. The time it triples is
A. 3
B. 4
C. 2(ln(3) − ln(2))
ln(2)
D. 3 ln(3)
E. 2 ln(3)
ln(2)
6
11. (5 points) Which of the following initial data will have a solution approaches +∞ as
t → +∞
dy
= ey (y − 1)(y − 2)3 (y − 3)2 , −∞ < y0 < ∞
dx
A. y(4) = 1
B. y(3) = 2
C. y(2) = 3
D. y(1) = 4
E. y(0) = 0
12. (7 points) Initially a tank holds 50 gallons of pure water. A salt solution containing 13 lb
of salt per gallon runs into the tank at the rate of 6 gallons per minute. The well mixed
solution runs out of the tank at a rate of 3 gallons per minute. Let x(t) be the amount
of salt in the tank at time t.
(a) The differential equation satisfied by x(t) is
(b) x(0) =
(c) Find the amount of salt, to the nearest pound, in the tank after 5 min
Answer: (a) x0 = 2 −
3x
;
50+3t
(b) x(0) = 0; (c) 9
7
13. (6 points)
(i) Solve for all a,
u0 (t) = |u(t)(u(t) − 1)|,
u(0) = a.
(ii) Draw (sketch) the solution with u(0) = 2.
Answer: (i) a = 0 : u(t) = 0; a = 1 : u(t) = 1; 0 < a < 1 : u(t) =
a
a−(a−1)et
8
et
; other
et + 1−a
a
a : u(t) =