What if a differential equation's right-hand side is an integer to a power of the dependent variable? What forms of guesses are appropriate?
It depends on what's on the left hand side.
For a second order DE with constant coefficients, you might try $\displaystyle y_p = a^x$, but this will not always work. Sometimes you might need $\displaystyle y_p = x a^x$ or even $\displaystyle y_p = x^2 a^x$ .....
Post the DE that inspired your question.