1. Find the general solution of y
0
= 3 y
4
− 8 xy
4
2. Solve the initial value problem y
00
+ 4 y
0
+ 13 y = 0 y (0) = 2 y
0
(0) = 1
3. Use the substitution z = ln y to find the general solution of the equation y
0
+ e x y = y ln y
4. Find the general solution of y
00
+ 2 y
0
+ y = t
2 − cos 4 t
5. Use the variation of parameters method to find a solution of y
00
+ 4 y = sec 2 t
6. Let
 1 0 0 
A =

6 1 1
0 4 − 2

(a) Find the eigenvalues and eigenvectors of A .
(b) Find the general solution of x
0
= A x .
7. Let
A =
1 − 1
5 − 3
(a) Find a fundamental matrix for x
0
= A x .
(b) Find the solution of x 0 = A x satisfying the initial condition x (0) =
1
− 1
8. The system x
0 y
0
= y
= y − x has (0 , 0) as its only equilibrium point. Classify this equilibrium point by stability and type, and sketch the phase portrait near (0 , 0).
9. Let f ( t ) =
(
2 0 < t < 3
0 t > 3
(a) Find the Laplace transform of f .
(b) Use the Laplace transform method to solve y
00 − 9 y = f ( t ) y (0) = 0 y
0
(0) = 1
10. We seek a solution of y
00
+ x
2 y
0
− 3 y = 0 in the power series form y ( x ) = P
∞ n =0 a n x n .
(a) What is the recurrence relation for the coefficients?
(b) If y (0) = 2 and y
0
(0) = 0 find a n for n = 0 , 1 , 2 , 3.