 Solution

With $z=(D-\beta )y$ and $\gamma =\alpha$, the first equation is $z^{\prime }-\alpha z=k{e}^{\alpha x}$. Multiplication by its integrating factor gives $(e^{-\alpha x}z{)}^{\prime }=k$, henceThis holds whether or not $\beta =\alpha$. If $\beta \ne \alpha$, solving $y^{\prime }-\beta y=z$ by a second integrating factor givesThe extra factor $x$ is the response to the resonance condition at the forcing exponent; the coincident-root case requires the next part.