Page 327 - Advanced engineering mathematics

10.2 Solution of X = AX for Constant A 307

Linear combination of these solutions are also solutions. Thus, define two solutions
1
( 1 (t) + 2 (t))
2
and
1
( 1 (t) − 2 (t)),
2i
and these are the solutions given in the theorem.

Theorem 10.7 enables us to replace two complex solutions

e (α+iβ) (U + iV) and e (α−iβ) (U − iV)
in the general solution with the two solutions given in the theorem, which involve only real
quantities.

EXAMPLE 10.9
We will solve the system X = AX with

⎛ ⎞
2 0 1
A = 0 −2 −2 ⎠ .
⎝
0 2 0
√ √
The eigenvalues are 2,−1 + 3i,−1 − 3i. Corresponding eigenvectors are, respectively,
⎛ ⎞ ⎛ ⎞ ⎛ ⎞
1 1 1
√ √
, and ⎝ 2 3i .
⎝ 0 ⎠ ⎝ −2 3i ⎠ ⎠
√ √
0 −3 + 3i −3 − 3i
One solution is
⎛ ⎞
1
2t
e .
⎝ 0 ⎠
0
Two other solutions are complex:
⎛ ⎞ ⎛ ⎞
1 1
√ √ √ √
e (−1+ 3i)t and ⎝ 2 3i e (−1− 3i)t .
⎝ −2 3i ⎠ ⎠
√ √
−3 + 3i −3 − 3i
These three solutions can be used as columns of a fundamental matrix. However, if we wish, we
can write a solution involving only real numbers and real-valued functions. First,
1 1 0
⎛ ⎞ ⎛ ⎞ ⎛ ⎞
√ √
= + i = U + iV
⎝ −2 3i ⎠ ⎝ 0 ⎠ ⎝ −2 3 ⎠
√ √
−3 + 3i −3 3
with
⎛ ⎞ ⎛ ⎞
1 0
√
U = ⎝ 0 ⎠ and V = −2 3 ⎠ .
⎝
√
−3 3
√
By Theorem 10.7, with α =−1 and β = 3, we can replace the two complex solutions with the
two solutions
√ √
−t
e [cos( 3t)U − sin( 3t)V]
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October 14, 2010 20:32 THM/NEIL Page-307 27410_10_ch10_p295-342