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:
$y'' + 8y' +15y=2e^{-3t}$

A good first guess for a solution would be: $Y(t) = Ae^{-3t}$.  However, there is a problem with this – when you try to find the constant $A$, everything cancels out on the left side and you end up with something like:  $0 = 2e^{-3t}$.

Why is this?  It’s exactly because your  guess, $Y(t) = Ae^{-3t}$happens to be a solution to the homogeneous equation:
$y'' + 8y' +15y=0$

When this occurs, you need to adjust your guess – we do that by multiplying by $t$, so a correct guess in this case should be:
$Y(t) = Ate^{-3t}$

If this guess also fails to work (as might happen in the case of a repeated root for the homogeneous equation), multiply by $t$ again:  $Y(t) = At^2e^{-3t}$

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