Page 123 - Schaum's Outline of Differential Equations

106 VARIATION OF PARAMETERS [CHAP. 12

Then, combining like terms, we have

12.3. Solve .

Here

Since y^ = ^,y^ = xe*, and <j>(x) = e"lx, it follows from Eq. (12.6) that

Solving this set of equations simultaneously, we obtain v{ = -1 and v 2 = IIx. Thus,

Substituting these values into (1), we obtain

The general solution is therefore,

12.4. Solve y"-y'-2y = e .
3x
Here

x
3
2
Since y 1 = e , y 2 = e *, and <j>(x) = e *, it follows from Eq. (12.6) that

4x
4x
Solving this set of equations simultaneously, we obtain v[ = -e /3 and V2 = e*l3, from which V 1 = -e /l2 and
x
v 2 = e /3. Substituting these results into (_/), we obtain

The general solution is, therefore,

(Compare with Problem 11.2.)