= Exponential forcing response with zero initial data
{title2=$(D-a)(D-b)y=e^{cx},\quad y(0)=y'(0)=0$}
For pairwise distinct rates $a,b,c$, the response is
$$
y=\frac{e^{ax}}{(a-b)(a-c)}+\frac{e^{bx}}{(b-a)(b-c)}+\frac{e^{cx}}{(c-a)(c-b)}.
$$
Set $v=y'-by$. Solving $v'-av=e^{cx}$ with $v(0)=0$ gives $(e^{cx}-e^{ax})/(c-a)$; solving the remaining first-order equation by an <integrating factor> gives the formula. Equivalently this is the convolution of $e^{cx}$ with the zero-displacement unit-impulse response $(e^{ax}-e^{bx})/(a-b)$ on the positive half-line. In the resonant case $c=b\ne a$, the finite limit is $y=[e^{ax}-e^{bx}-(a-b)xe^{bx}]/(a-b)^2$. The apparent singularity of the distinct-rate expression is removable, and the limit exhibits the polynomial factor of a <resonant exponential particular solution>.
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