Page 346 - Advanced engineering mathematics

326 CHAPTER 10 Systems of Linear Differential Equations

The matrix
⎛ ⎞
0 5 10
P = 1 0 1 ⎠
⎝
1 −4 −9
diagonalizes A. To make the change of variables i = PZ, we will need
⎛ ⎞ ⎛ ⎞
4 5 5 0
1
−1 ⎝ 10 −1
P = −10 10 ⎠ and P G = ⎝ 18 ⎠ .
10
−4 5 −5 −8
Now set i = PZ in the system to obtain
PZ = (AP)Z + G.

Multiply this system on the left by P −1 for
−1
−1
Z = (P AP)Z + P G

or
−1

Z = DZ + P G
in which D is the 3×3 diagonal matrix having the eigenvalues of A down its main diagonal. This
uncoupled system is
⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞
z 0 0 0 z 1 0
1

⎝ z ⎠ = 0 −2 0 ⎠⎝ z 2 ⎠ + ⎝ 18 ⎠ .
⎝
2
z 0 0 −20/9 z 3 −8
3
The uncoupled differential equations for the z j ’s are

z = 0
1
z + 2z 2 = 18

2
and
20

z + z 3 =−8,
3
9
which we solve individually to obtain
z 1 = c 1
z 2 = c 2 e −2t + 9
18
z 3 = c 3 e −20t/9 − .
5
Then
⎛ ⎞ ⎛ ⎞⎛ ⎞
0 5 10
i 1 z 1
= PZ = 1 0
i = i 2 ⎝ 1 ⎠⎝ z 2 ⎠
⎝ ⎠
1 −4 −9
i 3 z 3
⎛ ⎞⎛ ⎞
0 5 10 c 1
= 1 0 1 ⎠⎝ c 2 e −2t + 9 ⎠
⎝
1 −4 −9 c 3 e −20t/9 − 18/5
⎛ −2t −20t/9 ⎞
9 + 5c 2 e + 10c 3 e
= ⎝ c 1 + c 3 e −20t/9 − 18/5 ⎠ .
c 1 − 4c 2 e −2t − 9c 3 e −20t/9 − 18/5

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October 14, 2010 20:32 THM/NEIL Page-326 27410_10_ch10_p295-342