50 pts.
120 pts.
40 pts.
Problem 1 . Consider the autonomous differential equation dy
= 1 − y
2 dt
A. Sketch the graph of 1 − y 2 and positive.
to find out where this function is negative, zero
B. Use this information to sketch the phase portrait for the differential equation on the y -axis. Indicate the equilibrium solutions. Indicate if each equilibrium point is an attractor, a repellor, or a semi-stable point.
C. Use the information in the last part to sketch the graphs in the ty -plane of typical solutions of the differential equation in the regions between the equilibrium solutions.
Problem 2 . In each part, find the general solution of the differential equation.
If an initial condition is given, solve the initial value problem.
A.
dy dx
= 3 x
2 y
2
, y (1) = 1 .
B.
dy dx
= 2 x ( y + 1) y (0) = 3 .
C.
dy dx
+ 2 y = x
3 e
− 2 x
, y (0) = 1 .
D.
dy dx
−
2 x y = x
4
.
E.
dy dx
− 2 y = e x y
1 / 2
F.
dy dx
= ( x + y + 1)
2
Problem 3 . The following equation is exact. Solve it.
( y
3
+ 2 xy
2
+ 2 x ) dx + (3 xy
2
+ 2 x
2 y + y
2
) dy = 0 .
1

40 pts.
40 pts.
60 pts.
Problem 4 . Find an integrating factor that is a function of one of the variables alone, and use it to find the general solution of the following equation.
(2 xy
3
+ 2 x
2 y
3
) dx + 3 x
2 y
2 dy = 0 .
Problem 5 . The following equation is homogenous. Find the general solution.
( x
2
+ 2 y
2
) dx − xy dy = 0
Problem 6 . Newton’s law of cooling says that the time rate of change dT /dt of the temperature T of a body is proportional to the difference between T and the temperature T
M of the surrounding medium (the temperature of the surrounding medium is assumed to stay constant).
A cup of coffee at a temperature of 160
◦ is placed in a room that is at 70
◦
.
After 5 minutes the can has cooled to a temperature of 140
◦
.
A. Find the differential equation for the temperature T of the cup of coffee and solve it to find T as a function of time.
B. What will be the temperature of the coffee be after 10 minuites? Give a numerical answer accurate to two decimal places
C. At what time will the temperature of the coffee be 72
◦
? Give an numerical answer that is accurate to two decimal places.
2

EXAM
Exam 1
Math 3350, Summer II, 2010
July 20, 2010
• Write all of your answers on separate sheets of paper.
You can keep the exam questions when you leave.
You may leave when finished.
• You must show enough work to justify your answers.
approximations (e.g., 2, not 1 .
414).
• This exam has 6 problems. There are 350 points total .
Good luck!