As a rule, when finding the particular solution to a second order ODE, the trial solution has to consist of terms different from the ones from the corresponding homogeneous solution. Why should that be true? Assume we have found the correct form of the trial solution; why would adding a multiple from a fundamental solution of the homogeneous solution to the trial solution invalidate it? Given that we sum the homogeneous and particular solutions to find the general solution, the coefficients of the fundamental solutions should be able to offset the additional term in the particular solution. Moreover, why should the homogeneous solution have an effect on the inhomogeneous ODE?
When solving a few problems with such modified trial solutions it appears that the terms corresponding to the homogenous solution cancel each other nicely, leaving the particular solution unchanged. It therefore seems that this does not invalidate the particular solution (at least in some cases - if always, is there a proof?), it only renders the additional terms as redundant. Example: $$y''+3y'+2y=7.$$ The homogenous solution is $$y_H=Ae^{-t}+Be^{-2t}.$$ The trial solution is: $$y_P=C$$ which gives $y_P=\frac{7}{2}$. However, the trial solution $$y_P=C+5e^{-2t}$$ works equally well, giving the exact same particular solution. Same thing happens even with $y_P=C+y_H$.