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312 CHAPTER 10 Systems of Linear Differential Equations

Repeat this procedure for each eigenvalue of multiplicity greater than 1. Each eigenvalue
must have associated with it as many linearly independent solutions as the multiplicity of the
eigenvalue. This process terminates when n linearly independent solutions have been generated.

SECTION 10.2 PROBLEMS

In each of Problems 1 through 10, find a fundamental 13. 3 −5
matrix for the system and write the general solution as a 1 −1
matrix. If initial values are given, solve the initial value ⎛ 1 −1 1 ⎞
problem. 14. ⎝ 1 −1 0 ⎠
1 0 −1

1. x = 3x 1 , x = 5x 1 − 4x 2 ⎛ ⎞
1 2 −2 1 0

2. x = 4x 1 + 2x 2 , x = 3x 1 + 3x 2 15. ⎝ −5 0 0 ⎠
1 2
0 3 −2

3. x = x 1 + x 2 , x = x 1 + x 2
1 2
In each of Problems 16 through 21, find a fundamental

4. x = 2x 1 + x 2 − 2x 3 , x = 3x 1 − 2x 2 ,
1 2
matrix for the system with the given coefficient matrix.
x = 3x 1 − x 2 − 3x 3
3

2 0
5. x = x 1 + 2x 2 + x 3 , x = 6x 1 − x 2 , x =−x 1 − 2x 2 − x 3 16.
2
3
1
5 2
6. x = 3x 1 − 4x 2 , x = 2x 1 − 3x 2 ; x 1 (0) = 7, x 2 (0) = 5

1
2

3 2
7. x = x 1 − 2x 2 , x =−6x 1 ; x 1 (0) = 1, x 2 (0) =−19 17. 0 3

2
1
8. x = 2x 1 − 10x 2 , x =−x 1 − x 2 ; x 1 (0) =−3, x 2 (0) = 6 ⎛ 1 5 0 ⎞

2
1

9. x = 3x 1 − x 2 + x 3 , x = x 1 + x 2 − x 3 , x = x 1 − x 2 + 18. ⎝ 0 1 0 ⎠
3
1
2
x 3 ; x 1 (0) = 1, x 2 (0) = 5, x 3 (0) = 1 4 8 1
⎛ ⎞
10. x = 2x 1 + x 2 − x 3 , x = 3x 1 − 2x 2 , 2 5 6

1
2
x = 3x 1 + x 2 − 3x 3 ; x 1 (0) = 1, x 2 (0) = 7, x 3 (0) = 3 19. ⎝ 0 8 9 ⎠

3
0 1 −2
In each of Problems 11 through 15, find a real-valued fun- ⎛ 0 1 0 0 ⎞
damental matrix for the system X = AX with the given ⎜ 0 0 1 0 ⎟

⎟
coefficient matrix. 20. ⎜ 0 0 1 ⎠
⎝ 0
−1 −2 0 0

2 −4
11. ⎛ 1 5 −2 6 ⎞
1 2
0 3 0 4
⎜ ⎟
21.
⎜ ⎟
0 5 ⎝ 0 3 0 4 ⎠
12.
−1 −2 0 0 0 1
10.3 Solution of X = AX + G

We know that the general solution is the sum of the general solution of the homogeneous problem
X =AX plus any particular solution of the nonhomogeneous system. We therefore need a method

for finding a particular solution of the nonhomogeneous system. We will develop two methods.
10.3.1 Variation of Parameters
Variation of parameters for systems follows the same line of reasoning as variation of parameters
for second order linear differential equations. If (t) is a fundamental matrix for the homo-
geneous system X = AX, then the general solution of the homogeneous system is C.Using

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October 14, 2010 20:32 THM/NEIL Page-312 27410_10_ch10_p295-342