Find inverse Laplace transform L^{-1}left{frac{s-5}{s^2+5s-24}right} Please provide supporting details for your answer

Question
Laplace transform
asked 2020-12-27
Find inverse Laplace transform \(L^{-1}\left\{\frac{s-5}{s^2+5s-24}\right\}\) Please provide supporting details for your answer

Answers (1)

2020-12-28
Step 1
The following formulae are used in solving the given problem.
\(L^{-1}{f(x)+g(x)}=L^{-1}(f(x))+L^{-1}(g(x))\)
\(L^{-1}(a \cdot f(x))=a L^{-1}(f(x))\)
\(L^{-1}\left(\frac{1}{s+a}\right)=e^{-at}\)
Step 2
Evaluate the inverse Laplace transform of the given function as follows.
\(L^{-1}\left\{\frac{s-5}{s^2+5s-24}\right\}=L^{-1}\left\{-\frac{2}{11}(s-3)+\frac{13}{11}(s+8)\right\} (\text{ by partial fractions })\)
\(L^{-1}\left\{-\frac{2}{11(s-3)}\right\}+L^{-1}\left\{\frac{13}{11(s+8)}\right\}\)
\(-\frac{2}{11}L^{-1}\left\{1(s-3)\right\}+\frac{13}{11}L^{-1}\left\{\frac{1}{s+8}\right\}\)
\(-\frac{2}{11}e^{3t}+\frac{13}{11}e^{-8t}\)
Therefore, the inverse Laplace transform of the given function is \(-\frac{2}{11}e^{3t}+\frac{13}{11}e^{-8t}\)
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