Differential Equations Exam #1
Spring 2004
Name ________________________________
You must show all your work neatly and in numerical order on the paper provided. Solutions without correct supporting
work will not be accepted. Please do not write on the backs of your pages. Unless indicated otherwise the problem is
worth 12 points.
NO CALCULATORS MAY BE USED ON THIS EXAM.
1. Give the order of the differential equation and classify the equation as either linear or nonlinear.
a.
(4 points each)
sin  y  cos y  2
d 2 u du
b.

 u  cosr  u 
dr 2 dr
x  c1 cos t  c2 sin t is a two-parameter family of solutions of x  x  0 to find a solution of the
initial-value problem consisting of the differential equation and the initial conditions x / 6  1 / 2, x / 6  0 .
2. Use the fact that
(8 points)
3. Solve:
x
4. Solve:
x2
2
4
 dy
 2 x  8 xy
dx
dy
 y  xy
dx
y  5 y  y 2
5. Solve the Bernoulli equation:
6. Solve:
y
2

 yx dx  x 2 dy  0
7. Solve the “almost exact” equation:


6 xy dx  4 y  9 x 2 dy  0
8. A large tank is filled with 500 gallons of pure water. Brine containing 2 pounds of salt per gallon is pumped into the
tank at a rate of 10 gal/min. The well-mixed solution is pumped out of tank at a rate of 5 gal/min. Find the number
A(t ) of pounds of salt in the tank at time t .
9. Use reduction of order to find the general solution of the differential equation y   9 y  0 given that
solution.
y1  sin 3x is a