Problem Description:

Solve the following 2^nd order equation using the variation of parameter method:

Problem Solution:

1. The homogeneous solution for this problem was derived previously (see Example 2.1) as

2. Now for the particular solution, assume that y_p(x) can be written as

where

From our previous development for the Variation of Parameter technique (note that the system is already in standard form), we have

where

and

For performing the integrals and various manipulations, note the following integral relationships and trigonometric identities:

and

Now, with these expressions and relationships, the integrands and actual integrals become:

Therefore, y_p(x) becomes

or

Therefore, using the last trigonometric identity given above, one final simplification gives

Finally, combining this with the homogeneous solution gives

Clearly, Examples 2.1 and 2.2 show that, if possible, the method of undetermined coefficients should be used, since its application is usually much easier than the current variation of parameters approach. A word of advice is "Use the Variation of Parameters method when necessary, but only when necessary!"