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LINEAR SYSTEMS SOLUTION OF X = AX

FOR CONSTANT A SOLUTION OF
CHAPTER 10 X = AX + G EXPONENTIAL MATRIX

SOLUTIONS APPLICATIONS
Systems of

Linear

Differential

Equations

10.1 Linear Systems

We will apply matrices to the solution of a system of n linear differential equations in n unknown
functions:

x (t) = a 11 (t)x 1 (t) + a 12 (t)x 2 (t) + ··· + a 1n (t)x n (t) + g 1 (t)
1

x (t) = a 21 (t)x 1 (t) + a 22 (t)x 2 (t) + ··· + a 2n (t)x n (t) + g 2 (t)
2
.
.
.

x (t) = a n1 (t)x 1 (t) + a n2 (t)x 2 (t) + ··· + a nn (t)x n (t) + g n (t).
n
The functions a ij (t) are continuous and g j (t) are piecewise continuous on some interval (perhaps
the whole real line). Define matrices
⎛ ⎞ ⎛ ⎞
x 1 (t) g 1 (t)
x 2 (t) g 2 (t)
⎜ ⎟ ⎜ ⎟
A(t) =[a ij (t)],X(t) = ⎜ . ⎟ and G(t) = ⎜ . ⎟.
⎜
⎟
⎟
⎜
. .
⎝ . ⎠ ⎝ . ⎠
x n (t) g n (t)
Differentiate a matrix by differentiating each element. Matrix differentiation follows the usual
rules of calculus. The derivative of a sum is the sum of the derivatives, and the product rule has
the same form, whenever the product is defined:

(WN) = W N + WN .
With this notation, the system of linear differential equations is

X (t) = A(t)X(t) + G(t)
or

X = AX + G. (10.1)
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October 14, 2010 20:32 THM/NEIL Page-295 27410_10_ch10_p295-342