Differential Equations Exam #1
September 18, 2003
Name ________________________________
Show all your work neatly and in numerical order on notebook paper.
DO NOT WRITE ON THE BACKS OF YOUR PAGES.
1. Given
x  x  0 : (a) Show that x  c1 cos t  c2 sin t is a two-parameter family of solutions for the equation.
(b) Find a particular solution satisfying the initial conditions
2. Solve:
y
3. Solve:
2x  y  1y  1
4. Solve:
x  1 dy  y  ln x
2
x( / 2)  0, x / 2  1 .

 1 dx  y sec 2 x dy
dx
5. Solve the exact equation:
subject to the initial condition y (1)  10
5x  4 y dx  4 x  8 y 3 dy  0
6. Solve the Bernoulli equation:
dy
 y  ex y2
dx
7. The population of bacteria in a culture grows at a rate proportional to the number of bacteria present at time t. After 3
hours it is observed that there are 400 bacteria present. After 10 hours there are 2000 bacteria present. What was
the initial number of bacteria? You must start by writing a differential equation and solving it.
8. A large tank is partially filled with 100 gallons of fluid in which 10 pounds of salt is dissolved. Brine containing ½
pound of salt per gallon is pumped into the tank at a rate of 6 gal/min. The well-mixed solution is then pumped out at
a slower rate of 4 gal/min. Find the number of pounds of salt in the tank after 30 minutes.
9. A tank in the form of a right-circular cone standing on end, vertex down, is leaking water through a circular hole in its
bottom. Suppose the tank is 20 feet high and has radius 8 feet and the circular hole has radius 2 inches. The
dh
5
  3 / 2 . In this model, friction
dt
6h
g

32 ft / s 2 . If the tank is initially
and contraction of the water at the hole were taken into account with c  0.6 and
differential equation governing the height
h of water leaking from the tank is
full, how long will it take the tank to empty?