Question

The inverse Laplace transform of F(s) = $\frac{3s+5}{s^2+7}$ is _____________

(a) 3 sin ($\sqrt{7}$t) – $\frac{5}{\sqrt{7}}$  cos ($\sqrt{7}$ t)

(b) 3 sin ($\sqrt{7}$t) + $\frac{5}{\sqrt{7}}$  cos ($\sqrt{7}$ t)

(c) 3 cos ($\sqrt{7}$t) + $\frac{5}{\sqrt{7}}$  sin ($\sqrt{7}$ t)

(d) 3 cos ($\sqrt{7}$t) – $\frac{5}{\sqrt{7}}$  sin ($\sqrt{7}$ t)

I got this question during an interview.

My query is from Problems on Initial and Final Value Theorem in chapter Application of the Laplace Transform in Circuit Analysis of Network Theory

Answer 1

The correct option is (c) 3 cos ($\sqrt{7}$t) + $\frac{5}{\sqrt{7}}$  sin ($\sqrt{7}$ t)

To elaborate: F (t) = L^-1{F(s)}

= L^-1$\Big\{\frac{3s}{s^2+7} + \frac{5}{s^2+7}\Big\}$

= L^-1$\Big\{\frac{3s}{s^2 + \sqrt{7}^2} + \frac{5}{\sqrt{7}} \frac{\sqrt{7}}{s^2 +\sqrt{7}^2}\Big\}$

 The inverse Laplace transform is = 3 cos ($\sqrt{7}$t) + $\frac{5}{\sqrt{7}}$  sin ($\sqrt{7}$ t).