More on nonhomogeneous equations and WeBWorK 9

Or:  What to do when everything cancels out, and I’m left with nothing….

I wanted to address a problem that shows up when solving nonhomogeneous equations.  Consider this problem, similar to those in WeBWorK 9:

A good first guess for a solution would be: .  However, there is a problem with this – when you try to find the constant , everything cancels out on the left side and you end up with something like:  .

Why is this?  It’s exactly because your  guess, , happens to be a solution to the homogeneous equation:

When this occurs, you need to adjust your guess – we do that by multiplying by , so a correct guess in this case should be:

If this guess also fails to work (as might happen in the case of a repeated root for the homogeneous equation), multiply by again:  

Let me know if you have any questions,
Prof. Reitz