# Question

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

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 $$-\frac{2}{11}e^{3t}+\frac{13}{11}e^{-8t}$$