Page 121 - Schaum's Outline of Differential Equations

104 VARIATION OF PARAMETERS [CHAP. 12

For the case n = 2, Eqs. (12.4) become

and for the case n = 1, we obtain the single equation

Since y^(x), y 2(x), ... , y n(x) are n linearly independent solutions of the same equation L(j) = 0, their
Wronskian is not zero (Theorem 8.3). This means that the system (12.4) has a nonzero determinant and can be
solved uniquely for v[(x), v 2(x), ... ,v' n(x).

SCOPE OF THE METHOD
The method of variation of parameters can be applied to all linear differential equations. It is therefore more
powerful than the method of undetermined coefficients, which is restricted to linear differential equations with
constant coefficients and particular forms of (j)(x). Nonetheless, in those cases where both methods are applicable,
the method of undetermined coefficients is usually the more efficient and, hence, preferable.
As a practical matter, the integration of v'(x) may be impossible to perform. In such an event, other methods
(in particular, numerical techniques) must be employed.

Solved Problems

12.1. Solve /" + /= sec x.
This is a third-order equation with

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

Here y 1=l,y 2 = cos x, y 3 = sin x, and <j>(x) = sec x, so (12.5) becomes

Solving this set of equations simultaneously, we obtain v( = sec x, v^ = —I, and v 3 = -tan x. Thus,

Substituting these values into (_/), we obtain

The general solution is therefore