Page 340 - Advanced engineering mathematics

320 CHAPTER 10 Systems of Linear Differential Equations

The system of two second-order equations translates to the following system in terms of
x 1 ,··· , x 4 :

x = y = x 3 ,
1 1

x = y = x 4 ,

2
2

x = y =−8y 1 + 2y 2 =−8x 1 + 2x 2 ,

1
3

x = y = 2y 1 − 5y 2 = 2x 1 − 5x 2 ,

4
2
and
x 1 (0) = 1, x 2 (0) =−1, x 3 (0) = x 4 (0) = 0.
This is the system X = AX with

⎛ ⎞
0 0 1 0
0 0 0 1
⎜ ⎟
A = ⎜ ⎟
⎝ −8 2 0 0 ⎠
2 −5 0 0
and
⎛ ⎞
1
−1
⎜ ⎟
X(0) = ⎜ ⎟ .
⎝ 0 ⎠
0
A has the characteristic equation
2
2
(λ + 4)(λ + 9) = 0
with eigenvalues ±2i and ±3i. Corresponding to the eigenvalues 2i and 3i, we find two
eigenvectors
⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞
1 0 2 0
2 0 −1 0
⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟
+ i and + i .
⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟
⎝ 0 ⎠ ⎝ 2 ⎠ ⎝ 0 ⎠ ⎝ 6 ⎠
0 4 0 −3
The complex conjugates of these eigenvectors are also eigenvectors corresponding to eigenvalues
−2i and −3i. However, we will not write these other two eigenvectors, because we will use The-
orem 10.8 to write the four linearly independent solutions involving only real-valued functions.
From the eigenvector for 2i, write the two solutions
⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞
1 0 1 0
2
0
0
2
⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟
⎜ ⎟ cos(2t) − ⎜ ⎟ sin(2t) and ⎜ ⎟ sin(2t) + ⎜ ⎟ cos(2t).
⎝ 0 ⎠ ⎝ 2 ⎠ ⎝ 0 ⎠ ⎝ 2 ⎠
0 4 0 4
From the eigenvector for 3i, write the two solutions
⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞
2 0 2 0
⎜ −1 ⎟ ⎜ 0 ⎟ ⎜ −1 ⎟ ⎜ 0 ⎟
⎜ ⎟ cos(3t) − ⎜ ⎟ sin(3t) and ⎜ ⎟ sin(3t) + ⎜ ⎟ cos(3t).
⎝ 0 ⎠ ⎝ 6 ⎠ ⎝ 0 ⎠ ⎝ 6 ⎠
0 −3 0 −3
Use these four linearly independent solutions as columns of the fundamental matrix
⎛ ⎞
cos(2t) sin(2t) 2cos(3t) 2sin(3t)
⎜ 2cos(2t) 2sin(2t) −cos(3t) −sin(3t) ⎟
(t) = ⎜ ⎟ .
⎝ −2sin(2t) 2cos(2t) −6sin(3t) 6cos(3t) ⎠
−4sin(2t) 4cos(2t) 3sin(3t) −3cos(3t)
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October 14, 2010 20:32 THM/NEIL Page-320 27410_10_ch10_p295-342