Question

Find the \(L^{-1} (\frac{1}{(s^2+4)(s^2+9)})\).

(a) \(\frac{1}{5} \left (\frac{sin⁡(2t)}{2}-\frac{sin⁡(t)}{3}\right )\)

(b) \(\frac{1}{5} \left (\frac{sin⁡(2t)}{2}+\frac{sin⁡(3t)}{3}\right )\)

(c) \(\frac{1}{5} \left (\frac{sin⁡(t)}{2}-\frac{sin⁡(3t)}{3}\right )\)

(d) \(\frac{1}{5} \left (\frac{sin⁡(2t)}{2}-\frac{sin⁡(3t)}{3}\right )\)

The question was asked in final exam.

I'm obligated to ask this question of General Properties of Inverse Laplace Transform in section Laplace Transform of Engineering Mathematics

Answer 1

Correct option is (d) \(\frac{1}{5} \left (\frac{sin⁡(2t)}{2}-\frac{sin⁡(3t)}{3}\right )\)

To elaborate: In the given question,

\(L^{-1} \left (\frac{1}{(s^2+4)(s^2+9)}\right)\)

=\(\frac{1}{5} L^{-1} \left (\frac{5}{(s^2+4)(s^2+9)}\right)\)

=\(\frac{1}{5} L^{-1} \left (\frac{(s^2+9)-(s^2+4)}{(s^2+4)(s^2+9)}\right)\)

=\(\frac{1}{5} L^{-1} \left (\frac{1}{(s^2+4)}\right )-\frac{1}{5} L^{-1} \left (\frac{1}{(s^2+9)}\right)\)

=\(\frac{1}{5} \left (\frac{sin⁡(2t)}{2}-\frac{sin⁡(3t)}{3}\right)\).