Question

Find the \(L^{-1} \left (\frac{s}{(s^2+1)(s^2+2)(s^2+3)}\right )\).

(a) \(\frac{1}{2} cos⁡(t)-cos⁡(\sqrt3t)-\frac{1}{2} cos⁡(\sqrt3t)\)

(b) \(\frac{1}{2} cos⁡(t)+cos⁡(\sqrt2t)-\frac{1}{2} cos⁡(\sqrt3t)\)

(c) \(\frac{1}{2} cos⁡(t)-cos⁡(\sqrt2t)-\frac{1}{2} cos⁡(\sqrt3t)\)

(d) \(\frac{1}{2} cos⁡(t)+cos⁡(\sqrt2t)+\frac{1}{2} cos⁡(\sqrt3t)\)

I have been asked this question in an online interview.

The doubt is from General Properties of Inverse Laplace Transform topic in chapter Laplace Transform of Engineering Mathematics

Answer 1

The correct answer is (c) \(\frac{1}{2} cos⁡(t)-cos⁡(\sqrt2t)-\frac{1}{2} cos⁡(\sqrt3t)\)

The explanation: In the given question,

\(L^{-1} \left (\frac{s}{(s^2+1)(s^2+2)(s^2+3)}\right )\)

=\(L^{-1} \left (\frac{\frac{1}{2}}{(s^2+1)}+\frac{(-1)}{(s^2+2)}+\frac{\frac{(-1)}{2}}{(s^2+3)}\right )\) ——————-By method of Partial fractions

=\(\frac{1}{2} cos⁡(t)-cos⁡(\sqrt2t)-\frac{1}{2} cos⁡(\sqrt3t)\).