Question

Find the laplace transform of y(t)=e^t.t.Sin(t)Cos(t).

(a) \(\frac{4(s-1)}{[(s-1)^2+4]^2}\)

(b) \(\frac{2(s+1)}{[(s+1)^2+4]^2}\)

(c) \(\frac{4(s+1)}{[(s+1)^2+4]^2}\)

(d) \(\frac{2(s-1)}{[(s-1)^2+4]^2}\)

I had been asked this question in an international level competition.

The doubt is from Laplace Transform by Properties topic in portion Laplace Transform of Engineering Mathematics

Answer 1

Correct option is (d) \(\frac{2(s-1)}{[(s-1)^2+4]^2}\)

For explanation: y(t)=\(\frac{1}{2} t.e^t Sin(2t)\)

Laplace transform of Sin(2t)=\(\frac{2}{s^2+4}\)

Laplace transform of tSin(2t)=\(-\frac{d}{dt} \frac{2}{s^2+4}=\frac{2(2s)}{(s^2+4)^2}=\frac{4s}{(s^2+4)^2}\)

Laplace transform of te^t Sin(2t)=\(\frac{4(s-1)}{[(s-1)^2+4]^2}\)

Laplace transform of 1/2 te^t Sin(2t)=\(\frac{2(s-1)}{[(s-1)^2+4]^2}\)