Page 166 - Schaum's Outline of Differential Equations

CHAP. 17] REDUCTION OF LINEAR DIFFERENTIAL EQUATIONS 149

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REDUCTION OF AN n -ORDER EQUATION
As in the case of the second-order differential equation, with associated initial conditions, we can recast
higher order initial-value problems into a first-order matrix system as illustrated below:

with b n(t) jt 0, can be reduced to the first-order matrix system

where A(?), f(f), c, and the initial time t 0 are known. The method of reduction is as follows.
Step 1. Rewrite (77.5) so that d"xldt" appears by itself. Thus,

where and

Step 2. Define n new variables (the same number as the order of the original differential equation);
Xi(t), x 2(t), ..., x n(t), by the equations

These new variables are interrelated by the equations

Step 3. Express dx nldt in terms of the new variables. Proceed by first differentiating the last equation of (17.9)
to obtain

Then, from Eqs. (77.8) and (77.9),

For convenience, we rewrite this last equation so that Xi(f), appears before x 2(f), etc. Thus,