Page 122 - Schaum's Outline of Differential Equations

CHAP. 12] VARIATION OF PARAMETERS 105

12.2. Solve

This is a third-order equation with

(see Chapter 10); it follows from Eq. (12.3) that

x
2x
Here y l =l,y 2 = e*, y 3 = e , and <j)(x) = e /(l + e *), so Eq. (12.5) becomes

Solving this set of equations simultaneously, we obtain

x
Thus, using the substitutions u = e* + 1 and w = 1 + e , we find that

Substituting these values into (1), we obtain

The general solution is

This solution can be simplified. We first note that

so