Engineering Signal and Systems: Verify the impulse response of this system

AI Thread Summary
The discussion focuses on verifying the impulse response of the system defined by the equation dy(t)/dt + 2y(t) = x(t), with the proposed impulse response h(t) = e^{-2t}u(t). The initial attempt to verify this resulted in an incorrect simplification, leading to the erroneous conclusion that 0 = δ(t). The key mistake identified was neglecting the product rule for differentiation when handling the term e^{-2t}u(t). After applying the product rule correctly, the verification process was successfully completed. The final statement must reflect a true equality to confirm the impulse response.
SumDood_
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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?
 
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SumDood_ said:
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{d}{dt}[e^{-2t}u(t)] \neq -2e^{-2t}u(t) ##. Use the product rule, ##u(t)## is also a differentiable function of ##t##.
 
DaveE said:
## \frac{d}{dt}[e^{-2t}u(t)] \neq -2e^{-2t}u(t) ##. Use the product rule, ##u(t)## is also a differentiable function of ##t##.
Got the right solution, thanks!
 
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