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

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 $x{e}^{ax}$. The arbitrary constant must be reparametrized before taking the limit.