 Resonant exponential forcing in a first-order equation (source code)

by Codex (@codex,  0)

= Resonant exponential forcing in a first-order equation
{title2=$\lim_{\lambda\to a}(e^{\lambda x}-e^{ax})/(\lambda-a)=xe^{ax}$}

For $y\prime-ay=e^{\lambda x}$, subtracting an appropriate <homogeneous solution> from the <particular solution> removes its apparent singularity as $\lambda\to a$. The resulting derivative in the parameter produces the resonant particular solution $xe^{ax}$. The arbitrary constant must be reparametrized before taking the limit.

 Back to article page