Differential Equations Exam #3
Fall 2003
Name _________________________________
Show all your work neatly and in numerical order on notebook paper. You must omit one problem by clearly writing
“OMIT” by the problem on your notebook paper. If you do not omit a problem, I will omit the last one for you.
DO NOT WRITE ON THE BACKS OF YOUR PAGES.
1. A 12-pound weight stretches a spring 2 feet. The weight is released from a point 1 foot below the equilibrium position
with an upward velocity of 4 ft/s.
(a) Find the equation describing the resulting simple harmonic motion.
(b) What are the amplitude, period, and frequency of motion?
(c) At what times does the weight return to the point 1 foot below the equilibrium position?
(d) What is the velocity of the weight at t  3 /16 seconds?
(e) At what time is the velocity zero?
2. A beam of length L is embedded at the left end and free at its right end. Find the deflection of the beam if a constant
load w0 is uniformly distributed along its length – that is, w( x)  w0 , 0  x  L .
3. Use the definition of the Laplace transform to find
4. Evaluate:
a.
1


L1  2

 s  s  20 
5. Use the Laplace transform to solve:
6. Evaluate:
a.
 
L te4t .
b.
 se s / 2 
L1  2

s 4
y  4 y  6e3t  3et ,
L3t  1U (t  1)
b.

y(0)  1, y(0)  1
t
L  sin  cos(t   ) d
0
7. Solve:
t , 0  t  1
y  2 y  f (t ), y (0)  0, where f (t )  
0, t  1
8. Solve:
y  y   (t  2 ),

y (0)  0, y(0)  1
1 
 2
8 
1   et ,  2   et   tet , are solutions of a system   A . Determine whether the
  1
6
  8
vectors form a fundamental set on  ,  .
9. The vectors
10. Solve:
dx
 3x  2 y  4 z
dt
dy
 2x  2z
dt
dz
 4 x  2 y  3z
dt
11. Solve:
 4  5
 
  
 5  4