Non homogeneous differential equation - Particular solution

[mathmari]

Hey!

When we have the non-homogeneous differential equation $$ay''(x)+by'(x)+cy(x)=f(x)$$ and the non-homogeneous term $f(x)$ is of the form $e^{mx}P_n(x)$ we know that the particular solution is $$y_p=x^k(A_0+A_1x+ \dots +A_nx^n)e^{mx}$$ where $k$ is the multiplicity of the eigenvalue $w=m$. When we have the non-homogeneous differential equation $$ay''(x)+by'(x)+cy(x)=\sum_i s_ix^ie^{r_i x}$$ will the particular solution be $$y_p=\sum_i h_i x^{k_i}x^i e^{r_ix}$$ where $k_i$ is the multiplicity of $r_i$ ? (Wondering)Or isn't $\displaystyle{\sum_i s_ix^ie^{r_i x}}$ the general form of an element of the ring that contains the polynomials and the exponential sums? (Wondering)

 

[gtoguy97]

Yes, the particular solution to the non-homogeneous differential equation $$ay''(x)+by'(x)+cy(x)=\sum_i s_ix^ie^{r_i x}$$ will be $$y_p=\sum_i h_i x^{k_i}x^i e^{r_ix}$$ where $k_i$ is the multiplicity of $r_i$. $\displaystyle{\sum_i s_ix^ie^{r_i x}}$ is indeed the general form of an element of the ring that contains the polynomials and the exponential sums.