Find the inverse Laplace transform f(t)=L^)-1 {F(s)} of each of the following functions.

Question

Find the inverse Laplace transform $f (t) = L^{- 1} {F (s)}$ of each of the following functions.
$(i) F (s) = \frac{2 s + 1}{s^{2} - 2 s + 1}$
Hint – Use Partial Fraction Decomposition and the Table of Laplace Transforms.
$(i i) F (s) = \frac{3 s + 2}{s^{2} - 3 s + 2}$
Hint – Use Partial Fraction Decomposition and the Table of Laplace Transforms.
$(i i i) F (s) = \frac{3 s^{2} + 4}{(s^{2} + 1) (s - 1)}$
Hint – Use Partial Fraction Decomposition and the Table of Laplace Transforms.

Answer 1

$(i) F (s) = \frac{2 s + 1}{s^{2} - 2 s + 1}$
$\frac{2 s + 1}{{(s - 1)}^{2}} = \frac{A}{s - 1} + \frac{B}{{(s - 1)}^{2}}$
$2 s + 1 = \frac{A}{s - 1} + \frac{B}{{(s - 1)}^{2}}$
$2 s + 1 = A (s - 1) + B$
$A = 2, - A + B = 1$
$B = 3$
$F (s) = \frac{2}{s - 1} + \frac{3}{{(s - 1)}^{2}}$
$f (t) = 2 e^{t} + 3 t e^{t}$
$(i i) F (s) = \frac{3 s + 2}{s^{2} - 3 s + 2}$
$\frac{3 s + 2}{(s - 2) (s - 1)} = \frac{A}{s - 2} + \frac{B}{s - 1}$
$3 s + 2 = A (s - 1) + B (s - 2)$
$A + B = 3$
$A = 8, B = - 5$
$- A - 2 B = 2$
$F (s) = \frac{8}{s - 2} - \frac{5}{s - 1}$
$f (t) = 8 e^{2 t} - 5 e^{t}$
$(i i i) F (s) = \frac{3 s^{2} + 4}{(s^{2} + 1) (s - 1)}$
$\frac{3 s^{2} + 4}{(s - 1) (s^{2} + 1)} = \frac{A s + B}{s^{2} + 1} + \frac{C}{s - 1}$
$\Rightarrow 3 s^{2} + 4 = A (s^{2} - s) + B (s - 1) + C (s^{2} + 1)$
$A + B = 3$
$- A + B = 0$
$- B + C = 4$
$A = - \frac{1}{2}, B = - \frac{1}{2}, C = \frac{7}{2}$
$F (s) = \frac{- 1 s}{2 (s^{2} + 1)} - \frac{1}{2 (s^{2} + 1)} + \frac{7}{2 (s - 1)}$

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Find the inverse Laplace transform f(t)=L^)-1 {F(s)} of each of the following functions.

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