The GATE Grind

GATE 2023 ME – Question 36

Engineering Mathematics · Differential Equations: Initial and boundary value problems, Laplace transforms · 2 marks · Multiple choice

Which one of the options given is the inverse Laplace transform of $\frac{1}{s^3 - s}$? $u(t)$ denotes the unit-step function.

  1. $\left(-1 + \frac{1}{2}e^{-t} + \frac{1}{2}e^{t}\right)u(t)$
  2. $\left(\frac{1}{3}e^{-t} - e^{t}\right)u(t)$
  3. $\left(-1 + \frac{1}{2}e^{-(t-1)} + \frac{1}{2}e^{(t-1)}\right)u(t-1)$
  4. $\left(-1 - \frac{1}{2}e^{-(t-1)} - \frac{1}{2}e^{(t-1)}\right)u(t-1)$

Practise this question in The GATE Grind →

Show answer and explanation

Correct answer: (A) $\left(-1 + \frac{1}{2}e^{-t} + \frac{1}{2}e^{t}\right)u(t)$

Explanation

Factorise $s^3 - s = s(s - 1)(s + 1)$ and use partial fractions: $\frac{1}{s(s-1)(s+1)} = \frac{A}{s} + \frac{B}{s-1} + \frac{C}{s+1}$ with $A = \frac{1}{(-1)(1)} = -1$, $B = \frac{1}{(1)(2)} = \frac{1}{2}$ and $C = \frac{1}{(-1)(-2)} = \frac{1}{2}$. The inverse transform is $\left(-1 + \frac{1}{2}e^{t} + \frac{1}{2}e^{-t}\right)u(t)$.