Laplace II: Steps, Impulses & Convolution · Superchallenge problem

A damped oscillator obeys $y''+2y'+2y=\delta(t)+A\,\delta(t-T)$ with $y(0^-)=y'(0^-)=0$. For the smallest $T>0$ for which some $A$ makes $y\equiv0$ for all $t>T$, find that $A$.