= Solution
With $z=(D-\beta)y$ and $\gamma=\alpha$, the first equation is $z'-\alpha z=ke^{\alpha x}$. Multiplication by its <integrating factor> gives $(e^{-\alpha x}z)'=k$, hence
$$
\boxed{z=e^{\alpha x}(C+kx).}
$$
This holds whether or not $\beta=\alpha$. If $\beta\ne\alpha$, solving $y'-\beta y=z$ by a second <integrating factor> gives
$$
y=C_1e^{\alpha x}+C_2e^{\beta x}+\frac{kx e^{\alpha x}}{\alpha-\beta}.
$$
The extra factor $x$ is the response to the <resonance condition> at the forcing exponent; the coincident-root case requires the next part.
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