Page 124 - Schaum's Outline of Differential Equations

CHAP. 12] VARIATION OF PARAMETERS 107

2
12.5. Solve x + 4x = sin 2t.
This is a second-order equation for x(t) with

It follows from Eq. (12.3) that

and v 2 are now functions of t. Here ^ = cos 2t, x 2 = sin 2f are two linearly independent solutions of the
where v l
2
associated homogeneous differential equation and <j>(t) = sin 2f, so Eq. (12.6), with x replacing y, becomes

The solution of this set of equations is

Thus,

Substituting these values into (1), we obtain

2
2
because cos 2t + sin 2t = 1. The general solution is

12.6. Solve In? if it is known that two linearly independent solutions of the associated
homogeneous differential equation are t and t .
We first write the differential equation in standard form, with unity as the coefficient of the highest derivative.
f
Dividing the equation by , 2 we obtain

2
with <j>(t) = (lit) In t. We are given Ni = t and N 2 = t as two linearly independent solutions of the associated second-
order homogeneous equation. It follows from Theorem 8.2 that