Download + 2 5 2 3 2 2 x x x x x x − + = + − = − 2 3 24 2 14 7 5 6 x x x x x x x − +

Math 224 (Elementary Linear algebra and Differential Equations)
Review problems for Midterm 2
1. Find the general solution to the given differential equations:
a. y′′ −
c. y′′′
6y′ + 34y = 0
b. y′′
2
− 2y′′ − 4y′ + 8y = 0
e. y′′ −
+10y′ + 25y = 0
2
d. ( D + 2 D + 10) y = 0
y′ − 2y = 5e 2 x
f. y′′
+ 2y′ + 5y = 3sin 2x
2. Find an Annihilator for the given functions:
a. F(x) = 3 x e 2 x
b. F(x) = 10
c. F(x) = e − x + x 2
e. F(x) = 1 + 7x − x 2
f. F(x) = sin x + x
g. F(x) = sin 2 x
d. F(x) = − 6e3 x cos(5 x )
3. Solve y′′ − 6y′ + 34y = 0 subject to y (0) = 3 and y ′(0) = 4
 x1 − 2 x2 + 5 x3 = 2
using matrices [if infinite many solutions, clearly display them]

3x1 + 2 x2 − x3 = −2
4.
Solve
5.
2 x1 − x2 + 3 x3 = 24

Using the Gauss elimination method, solve 2 x2 − x3 = 14
7 x − 5 x = 6
2
 1
 0 6 2
1 4
 6 −1
 2 0 0
 −3 −2 








4  , D=
6. Given A = 4 1 0 , B = −1 5 , C = 2
 , and E =  0 −3 0  .





4 2
 −1 2 4 
 1 10 
 −3 5 
0 0 0








Find the followings (If possible)
a) 1/3 C
h)
E2
b) A B
c) B D
d) C B
i) Find the matrix X, if 2 X + 3 B = C
e) Trace of D
f)
BT
g) -5 B + 7 C
j) det(A) [By co-factor expansion method]
2 x1 − x2 − x3 = 0

7. Solve 5 x1 − x2 + 2 x3 = 0 using matrices. [If infinite many solutions, clearly display them]
x + x + 4x = 0
3
 1 2
Math 224 (Elementary Linear algebra and Differential Equations)
Review problems for Midterm 2
 x1 − 2 x2 + 4 x3 = 2

8. Solve 2 x1 − 3 x2 + 5 x3 = 3 For x3 only, using the Cramer’s-rule.
3x − 4 x + 7 x = 7
2
3
 1
9.
Verify that the vectors
1
 
x1 =  0  et ,
0
 
0
 −1
dx
  −t
 
x2 =  1  e , and x3 =  0  e 2t are solution to
= A x , where
dt
0
1
 
 
 1 0 −1


A =  0 −1 0 
0 0 2 


10. Determine the linear dependency or the linear independency of the following vectors:
a. {(2,−1),
(3, 2), (0,1)}
 1   1  0
     
b. { −1 , 2 , 0 }
     
 0   0   3
     
 1  2t  −2  −t
e , e }
 −1 
 2
c. {
11. Write a differential equation with the given characteristic roots and with the given multiplicities.
r1 = 2 ± 3i with multiplicity 2, r3 = −5 with multiplicity 1, and r4 = 2 with multiplicity 3.
Math 224 (Elementary Linear algebra and Differential Equations)
Review problems for Midterm 2