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!

  • $\begingroup$ What is your question? $\endgroup$ – user37238 Jul 9 '15 at 6:00
  • $\begingroup$ What should the limits of the integral be in the numerator? $\endgroup$ – 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 $$ y(t)=e^{-t}\int_0^ts\,e^s\,x(s)\,ds. $$

  • $\begingroup$ I don't understand why I should integrate between 0 and t. $\endgroup$ – user253368 Jul 9 '15 at 16:31
  • $\begingroup$ $t=0$ 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. $\endgroup$ – Julián Aguirre Jul 9 '15 at 16:53
  • $\begingroup$ ok that makes sense...now to find the impulse response simply make the input a delayed delta function? $\endgroup$ – user253368 Jul 9 '15 at 17:13

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