Exponential forcing response with zero initial data ($(D-a)(D-b)y=e^{cx},y(0)=y^{\prime }(0)=0$)

For pairwise distinct rates $a,b,c$, the response isSet $v=y^{\prime }-by$. Solving $v^{\prime }-av=e^{cx}$ 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 $y=[e^{ax}-e^{bx}-(a-b)x{e}^{bx}]/(a-b{)}^2$. The apparent singularity of the distinct-rate expression is removable, and the limit exhibits the polynomial factor of a resonant exponential particular solution.