MATH 308 EXAM 3 REVIEW
(1) Transform the given equation into a system of first order equations.
(a) y 00 − t3 y 0 + 5(t − 2)y = t5 − 2t3 + et , y(0) = 3, y 0 (0) = −2
(b) y (4) + 3y 000 − (t + 2)y 00 − 5y 0 + t2 y = cos t
(2) Either compute

2

1
(a) A =
−3

0

−3
(b) A =
2
the inverseof the given matrix, or else show that it is noninvertible (singular).
−2 4
−1 3 
1 −2

3 −6
6
9 
−3 −8
(3) Determine whether the given vectors are linearly independent. If they are linearly dependent, find
a linear relation
them.

 among
 
 
−1
1
2
(2)
(3)
(1)





2
0 ,x =
4 
,x =
(a) x =
−3
1
0




 
0
−2
3
(2)
(3)
(1)





2
−2 , x =
4 
,x =
(b) x =
−3
−1
0
(4) Solve the initial value problem
(
x01 = x1 + 3x2
x02 = 2x1 − 4x2
with the initial conditions x1 (0) = 4, x2 (0) = 5.
(5) Find the general
solution of

the system of first order equations.
3 −1 2
0

3 −1 6  x
(a) x =
−2 2 −2


5 −4 0
(b) x0 =  1 0 2  x
0 2 5
(
x01 = −4x1 − 3x2
(c)
x02 = 6x1 + 2x2
0
(d) x =
1 4
3 −3
x+
−3t
10e3t
1
2
(6) Find the general power series solution about the point x0 = 0 of the differential equation
2y 00 − 3xy 0 − 3y = 0
(7) Seek power series solutions of the given differential equation about the given point x0 . Find the
recurrence relation. Find the first four terms in each of two solutions y1 and y2 .
(a) y 00 − (2x + 1)y 0 + 2y = 0, x0 = 0
(b) xy 00 + 3xy 0 − y = 0,
x0 = 1
(8) Determine the indicial equation, the recurrence relation, and the roots of the indicial equation.
Find the general power series solution about x0 = 0 of the differential equation
3x2 y 00 + 8xy 0 − (x + 2)y = 0