Question

Find the \(L^{-1} (\frac{1}{s(s^2+4)})\).

(a) \(\frac{1-sin⁡(t)}{4}\)

(b) \(\frac{1-cos⁡(t)}{4}\)

(c) \(\frac{1-sin⁡(2t)}{4}\)

(d) \(\frac{1-cos⁡(2t)}{4}\)

I have been asked this question in an interview for job.

Enquiry is from Convolution in section Laplace Transform of Engineering Mathematics

Answer 1

Correct choice is (d) \(\frac{1-cos⁡(2t)}{4}\)

To elaborate: In the given question

Let p1(s) = \(\frac{1}{s^2+4}\) and p2(s) = \(\frac{1}{s}\)

\(f_1 (t) = L^{-1} (\frac{1}{s^2+4}) = \frac{sin⁡(2t)}{2}\)

\(f_2 (t) = L^{-1} (\frac{1}{s}) = 1\)

By Convolution Theorem,

\(L^{-1} (p_1(s)×p_2(s)) = \int_{0}^{t}f_1(u) f_2(t-u) dt\)

\(L^{-1} (\frac{1}{s(s^2+4)}) = \int_{0}^{t}\frac{1}{2} sin⁡(2u) du\)

=\(\frac{1-cos⁡(2t)}{4}\)

Thus, the answer is \(\frac{1-cos⁡(2t)}{4}\).