What is inverse oflaplace ((s+1)^2)/(s^2-s-6) ?

The answer to inverse oflaplace ((s+1)^2)/(s^2-s-6) is \delta(t)-1/5 e^{-2t}+16/5 e^{3t}

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inverse oflaplace ((s+1)^2)/(s^2-s-6)

inverse laplace

\frac{( s + 1 ) ^{2}}{s ^{2} - s - 6}

δ (t) - \frac{1}{5} e^{- 2 t} + \frac{16}{5} e^{3 t}

Solution steps

L^{- 1} {\frac{( s + 1 ) ^{2}}{s ^{2} - s - 6}}

Take the partial fraction of

\frac{( s + 1 ) ^{2}}{s ^{2} - s - 6} : 1 - \frac{1}{5 ( s + 2 )} + \frac{16}{5 ( s - 3 )}

= L^{- 1} {1 - \frac{1}{5 ( s + 2 )} + \frac{16}{5 ( s - 3 )}}

Use the linearity property of Inverse Laplace Transform:

For functions

f (s), g (s)

and constants

a, b : L^{- 1} {a \cdot f (s) + b \cdot g (s)} = a \cdot L^{- 1} {f (s)} + b \cdot L^{- 1} {g (s)}

= L^{- 1} {1} - L^{- 1} {\frac{1}{5 ( s + 2 )}} + L^{- 1} {\frac{16}{5 ( s - 3 )}}

L^{- 1} {1} : δ (t)

L^{- 1} {\frac{1}{5 ( s + 2 )}} : \frac{1}{5} e^{- 2 t}

L^{- 1} {\frac{16}{5 ( s - 3 )}} : \frac{16}{5} e^{3 t}

= δ (t) - \frac{1}{5} e^{- 2 t} + \frac{16}{5} e^{3 t}

What is inverse oflaplace ((s+1)^2)/(s^2-s-6) ?
The answer to inverse oflaplace ((s+1)^2)/(s^2-s-6) is \delta(t)-1/5 e^{-2t}+16/5 e^{3t}