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

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