Differential Equations Guessing

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