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So I'm trying to solve differential equations through the method of undetermined coefficients. I read this post which gave a good link to a table of guesses. That page says:

...the limitation of the method of undetermined coefficients is that the non-homogeneous term can only contain simple functions...

So does that mean your $g(t)$ can't be something like $sec^3(t)$? If my $g(t)=sec^3(t)$ then what other method can I use?

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You could try guessing $\sec^3 t$, but the trial function becomes more and more difficult the more complicated you guess becomes, which means that it is less likely to be a solution. –  Arkamis Mar 5 '13 at 20:02

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up vote 1 down vote accepted

As you say, it's a table of guesses, that work well in many cases ("most cases you'll see in practice" is probably the technical term ;-). That doesn't mean you can't come up with your own special cases.

The general solution to your quandary is the method of variation of parameters, you take the general solution to your homogeneous equation and replace the arbitrary constants in the solution by functions, plug the result into the inhomogeneous equation and solve for your functions.

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