Explained: "Find f(t) L−1{(3s−1)s2(s+1)3}"

Name: Find f(t) L−1{(3s−1)s2(s+1)3}
Uploaded: 2022-02-23T17:22:57+00:00
Description: Find f(t) L−1{(3s−1)s2(s+1)3}

Amari Flowers

Answered question

2021-09-18

Find f(t)
$L^{- 1} {\frac{(3 s - 1)}{s^{2} (s + 1)^{3}}}$

Answer & Explanation

Anonym

Skilled2021-09-19Added 108 answers

Step 1
From the given information it is needed to calculate:
$L^{- 1} {\frac{(3 s - 1)}{s^{2} (s + 1)^{3}}}$
Step 2
Now, first take the partial fractions of the given expression and solve:
$\frac{(3 s - 1)}{s^{2} {(s + 1)}^{3}} = \frac{a_{0}}{s} + \frac{a_{1}}{s^{2}} + \frac{a_{2}}{(s + 1)} + \frac{a_{3}}{{(s + 1)}^{2}} + \frac{a_{4}}{{(s + 1)}^{3}}$
$= \frac{a_{0} (s) {(s + 1)}^{3} + a_{1} {(s + 1)}^{3} + a_{2} (s^{2}) {(s + 1)}^{2} + a_{3} (s^{2}) (s + 1) + a_{4} (s^{2})}{s^{2} {(s + 1)}^{3}}$
$3 s - 1 = a_{0} (s) {(s + 1)}^{3} + a_{1} {(s + 1)}^{3} + a_{2} {(s + 1)}^{2} + a_{3} (s^{2}) (s + 1) + a_{4} (s^{2})$
when $s = 0 \Rightarrow 3 (0) - 1 = a_{1} {(1)}^{3} \Rightarrow a_{1} = - 1$
when $s = - 1 \Rightarrow 3 (- 1) - 1 = a_{4} {(- 1)}^{2} \Rightarrow - 4 = a_{4}$
Step 3
Now calculate rest of the unknowns by putting the value of the $a_{1} and a_{4}$ :
$3 s - 1 = a_{0} (s) {(s + 1)}^{3} + (- 1) {(s + 1)}^{3} + a_{2} (s^{2}) {(s + 1)}^{2} + a_{3} (s^{2}) (s + 1) + (- 4) (s^{2})$

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