Page 271 - Schaum's Outline of Differential Equations
P. 271
CHAPTER 26
Solutions of Linear
Differential Equations
with Constant
Coefficients by Matrix
Methods
SOLUTION OF THE INITIAL-VALUE PROBLEM
By the procedure of Chapter 17, any initial-value problem in which the differential equations are all linear
with constant coefficients, can be reduced to the matrix system
where A is a matrix of constants. The solution to Eq. (26.1) is
or equivalently
In particular, if the initial-value problem is homogeneous [i.e., f(t) = 0], then both equations (26.2) and (26.3)
reduce to
At
A< s)
(>
In the above solutions, the matrices e^ '°\e As , and e -' are easily computed from e by replacing the
variable t by t - t 0, -s, and t — s, respectively. Usually x(t) is obtained quicker from (26.3) than from (26.2),
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