Page 167 - Schaum's Outline of Differential Equations

150 REDUCTION OF LINEAR DIFFERENTIAL EQUATIONS [CHAP. 17

Step 4. Equations (17.10) and (17.11) are a system of first-order linear differential equations in x^t),
x 2(t), ...,x n(t). This system is equivalent to the single matrix equation ii(t) = A(?)x(?) + f(t) if we
define

Step 5. Define

Then the initial conditions (17.6) can be given by the matrix (vector) equation x(? 0) = c. This last equa-
tion is an immediate consequence of Eqs. (17.12), (17.13), and (17.6), since

Observe that if no initial conditions are prescribed, Steps 1 through 4 by themselves reduce any linear
differential Eq. (17.5) to the matrix equation x(t) = A.(t)x(t) + f(t).

REDUCTION OF A SYSTEM
A set of linear differential equations with initial conditions also can be reduced to System (17.7). The
procedure is nearly identical to the method for reducing a single equation to matrix form; only Step 2 changes.
With a system of equations, Step 2 is generalized so that new variables are defined for each of the unknown
functions in the set.