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,