# Differential Equations Exam #2 Spring 2004 Name_________________________________

```Differential Equations Exam #2
Spring 2004
Name_________________________________
Show all your work neatly and in numerical order on notebook paper.
DO NOT WRITE ON THE BACKS OF YOUR PAGES.
 c 2 e 3 x is the complementary function for y   y   6 y  g ( x) , write the form of the
particular solution for the given values of g (x ) . Do not solve.
1. Given that y c  c1e
a.
g ( x)  sin 4 x
b.
g ( x)  xe3 x
2 x
2. Solve:
y 4   y' ' ' y' '  0
3. Solve:
5 y   y   6 x
4. a. Find a linear differential operator that annihilates 13 x  9 x  sin 4 x .
2
3 5x
b. Find a linear differential operator that annihilates x e
5. Use variation of parameters to solve:
6. Solve:
.
y   3 y   2 y  sin e x
d2y
dy
4x
 8x  y  0
2
dx
dx
2
7. Solve the system:
Dx  D  2  y  0
D  3x  2 y  0
8. Solve using the substitution u  y  :
 y  1y    y 2
9. 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/sec.
a. Find the equation describing the resulting simple harmonic motion.
b. At what time does the weight return to the point 1 foot below equilibrium position?
10. Find the eigenvalues and eigenfunctions for y   y  0,
y (0)  0,
 
y   0 .
4
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