A voltage v(t)=2t^(2)e^(-3t) is passed through a delay line having a delay of 4 sec.Detrmine Laplace transform of the output voltage

Question

anonymous · Answer

I think your question was: $$v(t) = 2t ^{2} e ^{-3t}$$
You need to apply the following rules:
$$L[u(t)] = 1/s$$
$$L[e ^{at} f(t)] = F(s - a)$$
$$L[t ^{n} f(t)] = (-1)^{n}F ^{n}(s)$$
The Laplace transform is:$$ V(s) = 4/(s + 3)^{3}$$

Now to account for the delay. The time shift manifests itself as a phase shift in s domain. So, an additional e^(-as) term joins the result. So, the final result after 4s delay is:

$$ V(s) = (4 e^{-4s})/(s + 3)^{3}$$