EXAM
Exam 1
Math 3350, Summer 2011
June 13, 2001
• Write all of your answers on the blank paper
provided. Do not write on the exam questions
handout. When finished, write your name and the
section number on the first page of your answers.
Staple the exam questions and your sheet of notes to
the back of your answers. You may leave when
finished.
• You must show enough work to justify your answers.
Unless otherwise instructed,
give exact answers, not
√
approximations (e.g., 2, not 1.414).
• This exam has 5 problems. There are 310 points
total.
Good luck!
40 pts.
Problem 1. Consider the autonomous differential equation
dy
= y(y + 1)(y − 2).
dx
A. Find the constant solutions of this equation. Draw the phase portrait for
this equation.
B. In the xy-plane, sketch the graphs of typical solutions in the regions divided
by the constant solutions.
120 pts.
Problem 2. In each part, find the general solution of the differential equation,
or solve the given initial value problem. You must show the steps
in solving the equation by one of the methods given in class, you can’t just write
down the answer. Solve for y, if it’s easy to do.
A.
B.
C.
D.
E.
F.
40 pts.
dy
= 3x2 y 2 ,
dx
y(0) = 2
dy
= cos(x)y
dx
dy
+ 3y = e2x ,
dx
y(0) = −1
2
dy
+ y = x3
dx x
dy
+ 2y = e2x y 2
dx
dy
x2 + 2y 2
=
dx
xy
Problem 3. The following differential equation is exact. Find the general
solution.
(y 2 + 2xy 3 + 2x) dx + (2xy + 3x2 y 2 + 1) dy = 0
1
40 pts.
70 pts.
Problem 4. The following differential equation is not exact. Find an integrating factor that is a function of x alone, or an integrating
factor that is a function of y alone. Find the general solution of the differential
equation.
(2xy + x2 y) dx + x2 dy = 0
Problem 5. A can of soda at a temperature of 40 degrees (Fahrenheit) is
brought into a room with a room temperature of 70 degrees.
After 10 minutes, the temperature of the can is 50 degrees.
Recall that the differential equation for newton’s law of cooling is
dT
= k(T − TM ).
dt
Start by writing down the solution of this equation (you don’t need to show the
details of solving the equation).
A. Find the value of k for this problem. Give an exact answer, not an approximation.
B. Find the temperature of the can after 20 minutes. Give an exact answer and
an approximation to two decimal places.
C. At what time will the temperature of the can be 69 degrees? Give an exact
answer and an approximation to two decimal places.
2