Page 172 - Schaum's Outline of Differential Equations

CHAP. 17] REDUCTION OF LINEAR DIFFERENTIAL EQUATIONS 155

we define the three new variables

Then,

or,

These equations are equivalent to the matrix equation x(t) = A.(t)x(t) + f(t) if we define

If we also define t 0 = 0 and c = then the initial conditions can be given by x(t 0) = c.

17.9. Put the following system into matrix form:

We proceed exactly as in Problems 17.7 and 17.8, except that now there are no initial conditions to consider.
Since the system consists of two first-order differential equations, we define two new variables x 1(t)=x and
yi(f) = y. Thus,

If we define

then this last set of equations is equivalent to the matrix equation x(t) = A.(t)x(t) + f(t), or simply to x(t) = A.(t)x(t),
since f(t) = 0.

Supplementary Problems

Reduce each of the following systems to a first-order matrix system.

17.10. x-2x + x = t + l;x(l) = l,x(l) = 2

17.11. 2x + x = 4e';x(0) = l,x(0) = l
2
17.12. tx - 3x - t x = sin t; x(2) = 3, x(2) = 4