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[mathmari]

Hey!

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 $f=\alpha_ie^{b_i x}$, 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 $$y(x)=e^{b_i x} (A_0+A_1x+ \dots +A_{M}x^{M})$$ ? (Wondering)

 

[gtoguy97]

The particular solution in this case is the second one: $$y(x)=e^{b_i x} (A_0+A_1x+ \dots +A_{M}x^{M})$$. The first solution is only valid when $L$ has a constant coefficient.