Suppose we have a non-homogeneous differential equation $Ly=f$ in the ring of exponential sums $\mathbb{C}[e^{\lambda x} \mid \lambda \in \mathbb{C}]$, then $f=e^{b_i x}$, right?
When $b_i$ is a root of the characteristic equation of the homogeneous equation of multiplicity $M$, then how is the particular solution?
Is it $$y(x)=Cx^Me^{b_i x}$$ or $$y(x)=e^{b_i x} (A_0+A_1x+ \dots +A_{M}x^{M})$$ ?
Since $$ y(x)=e^{b_i x} (A_0+A_1x+ \dots +A_{M-1}x^{M-1})$$ solves the homogeneous equation $Ly = 0$, you can use either. But using the first form will probably be less work.