What is the impulse response of the system

Question

Given this input-output system what is the impulse response

()/dt + () = (), ≥ 0, (0) = 0

I used an integrating factor to find y(t)

y(t) = ${\int t*x(t) *e^tdt\over e^t }$

From here i thought I should use the replace x(t) with an impulse, but I'm not sure of what the limits should be for the integral. Thanks for any help!

Answer 1

When you multiply by the integrating factor you get $$ (e^t\,y)'=t\,e^t\,x $$ Integrate betwwen $0$ and $t$ to get $$ e^t\,y(t)-y(0)=\int_0^ts\,e^s\,x(s)\,ds $$ and $$ y(t)=e^{-t}\int_0^ts\,e^s\,x(s)\,ds. $$