Question

Obtain the inverse Laplace transform : \[\]$\dfrac{{2s - 1}}{{{s^3} - s}}$

Answer 1

Hint: To find the inverse Laplace form we use the following notation:
$f(t) = {L^{ - 1}}\left\{ {F(s)} \right\}$
In the following question ,$F(s) = \dfrac{{2s - 1}}{{{s^3} - s}}$

Complete step by step answer:
$f(t) = {L^{ - 1}}\left\{ {\dfrac{{2s - 1}}{{{s^3} - s}}} \right\}$
Now,
$
   = {L^{ - 1}}\left\{ {\dfrac{{2s - 1}}{{{s^3} - s}}} \right\} \\
   = {L^{ - 1}}\left\{ {\dfrac{{2s - 1}}{{s({s^2} - 1)}}} \right\} \\
   = {L^{ - 1}}\left\{ {\dfrac{{2s - 1}}{{s(s - 1)(s + 1)}}} \right\} \\
 $
Using partial fraction ;
$
  \dfrac{A}{s} + \dfrac{B}{{s - 1}} + \dfrac{C}{{s + 1}} = \dfrac{{2s - 1}}{{s(s - 1)(s + 1)}} \\
  \dfrac{{A(s - 1)(s + 1) + Bs(s + 1) + Cs(s - 1)}}{{s(s - 1)(s + 1)}} = \dfrac{{2s - 1}}{{s(s - 1)(s + 1)}} \\
  \dfrac{{A({s^2} - 1) + B({s^2} + s) + C({s^2} - s)}}{{s(s - 1)(s + 1)}} = = \dfrac{{2s - 1}}{{s(s - 1)(s + 1)}} \\
 $


Comparing the coefficients of ${s^2}$:
$A + B + C = 0 - (i)$
Comparing the coefficient of $s$:
$B - C = 2 - (ii)$
Comparing the coefficients of constants:
$
   - A = - 1 \\
  A = 1 - (iii) \\
 $
From $(i)$ and $(ii)$
$
  B = C + 2 \\
  A + B + C = 0 \\
  A + C + 2 + C = 0 \\
  A + 2C = - 2 \\
    \\
 $
From $(iii)$
$
  1 + 2c = - 2 \\
  2c = - 2 - 1 \\
  2c = - 3 \\
  c = \dfrac{{ - 3}}{2} \\
 $
Now,
$
  B = C + 2 \\
  B = \dfrac{{ - 3}}{2} + 2 \\
  B = \dfrac{1}{2} \\
 $
Hence;
$
   = {L^{ - 1}}\left\{ {\dfrac{1}{s} + \dfrac{1}{2}(\dfrac{1}{{s - 1}}) - \dfrac{3}{2}(\dfrac{1}{{s + 1}})} \right\} \\
   = {L^{ - 1}}\left\{ {\dfrac{1}{s}} \right\} + \dfrac{1}{2}{L^{ - 1}}\left\{ {\dfrac{1}{{s - 1}}} \right\} - \dfrac{3}{2}{L^{ - 1}}\left\{ {\dfrac{1}{{s + 1}}} \right\} \\
    \\
 $
Since, the standard formula is ${L^{ - 1}}\left\{ {\dfrac{1}{{s - a}}} \right\} = {e^{at}}$
$
   = {e^{0t}} + \dfrac{1}{2}{e^{1t}} - \dfrac{3}{2}{e^{ - 1t}} \\
   = 1 + \dfrac{1}{2}{e^t} - \dfrac{3}{2}{e^{ - t}} \\
 $

Note: Laplace transform is a mathematical tool mostly used in the engineering field to transform an equation from one form to another. It is also an integral transform that converts a function of a real variable (often time) to a function of a complex variable.