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scZoUnD [109]
3 years ago
10

Inverse laplace of L^-1 {5s/s^2 + 3s - 4}​

Mathematics
1 answer:
kap26 [50]3 years ago
4 0

Answer:

e^{t}+4e^{-4t}.

Step-by-step explanation:

To find the Laplace inverse of the above problem, we have to first solve the problem as a partial fraction, hence:

\frac{5s}{s^2+3s-4}=\frac{5s}{s^2+4s-s-4}  \\\\=\frac{5s}{s(s+4)-1(s+4)}\\\\=\frac{5s}{(s-1)(s+4)}=\frac{A}{s-1}+\frac{B}{s+4}\\\\=  \frac{A(s+4)+B(s-1)}{(s-1)(s+4)}\\\\Hence:\\\\\frac{A(s+4)+B(s-1)}{(s-1)(s+4)}=\frac{5s}{(s-1)(s+4)}\\\\A(s+4)+B(s-1)=5s\\\\to\ find \ A,put\ s=1:\\\\A(1+4)+B(1-1)=5(1)\\\\5A=5\\\\A=1\\\\to\ find \ B,put\ s=-4:\\\\A(-4+4)+B(-4-1)=5(-4)\\\\-5B=-20\\\\B=4\\\\Hence:\\\\\frac{5s}{s^2+3s-4}=\frac{1}{s-1}+ \frac{4}{s+4}\\\\

L^{-1} [\frac{5s}{s^2+3s-4}]=L^{-1}[\frac{1}{s-1} ]+L^{-1}[\frac{4}{s+4} ]\\\\But\ L^{-1}[\frac{1}{s-a} ]=e^{at}\\\\Hence:\\\\L^{-1} [\frac{5s}{s^2+3s-4}]=L^{-1}[\frac{1}{s-1} ]+L^{-1}[\frac{4}{s+4} ]=e^{t}+4e^{-4t}\\\\=e^{t}+4e^{-4t}

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