# What is the impulse response of the system

Given this input-output system what is the impulse response

𝑑𝑦(𝑡)/dt + 𝑦(𝑡) = 𝑡𝑥(𝑡), 𝑡 ≥ 0, 𝑦(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!

• What is your question? – user37238 Jul 9 '15 at 6:00
• What should the limits of the integral be in the numerator? – user253368 Jul 9 '15 at 6:10

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 $$y(t)=e^{-t}\int_0^ts\,e^s\,x(s)\,ds.$$
• $t=0$ is the starting time at which you have the initial condition $y(0)=0$. And $t$ is the time at which you want the solution. – Julián Aguirre Jul 9 '15 at 16:53