- #1

SumDood_

- 30

- 6

- Homework Statement
- Verify that the impulse response of this system is h(t) = e^(-2t)u(t)

- Relevant Equations
- Impulse response => x(t) = impulse

Verify that the impulse response of this system is ##h(t) = e^{-2t}u(t)## for the following system

$$\frac{dy(t)}{dt} + 2y(t) = x(t)$$

So this is what I did first,

$$

\begin{align}

\frac{dh(t)}{dt} + 2h(t) &= \delta (t) \\

\frac{d}{dt}e^{-2t}u(t) + 2e^{-2t}u(t) &= \delta (t) \\

-2e^{-2t}u(t) + 2e^{-2t}u(t) &= \delta (t) \\

0 &= \delta (t)

\end{align}

$$

Of course, my solution is wrong. Honestly, at the beginning, I didn't know what I am supposed to get that would verify that the impulse response of the system is ##h(t) = e^{-2t}u(t)##.

So, first, what mistake did I make when simplifying after substituting ##h(t)## and ##x(t)##?

Second, what form does the final statement need to take to actually verify the impulse response? Should I end up with a true statement?

$$\frac{dy(t)}{dt} + 2y(t) = x(t)$$

So this is what I did first,

$$

\begin{align}

\frac{dh(t)}{dt} + 2h(t) &= \delta (t) \\

\frac{d}{dt}e^{-2t}u(t) + 2e^{-2t}u(t) &= \delta (t) \\

-2e^{-2t}u(t) + 2e^{-2t}u(t) &= \delta (t) \\

0 &= \delta (t)

\end{align}

$$

Of course, my solution is wrong. Honestly, at the beginning, I didn't know what I am supposed to get that would verify that the impulse response of the system is ##h(t) = e^{-2t}u(t)##.

So, first, what mistake did I make when simplifying after substituting ##h(t)## and ##x(t)##?

Second, what form does the final statement need to take to actually verify the impulse response? Should I end up with a true statement?