What is inverse oflaplace (s+5)/((s+3)(s-1)) ?

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

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inverse oflaplace (s+5)/((s+3)(s-1))

inverse laplace

\frac{s + 5}{( s + 3 ) ( s - 1 )}

- \frac{1}{2} e^{- 3 t} + \frac{3}{2} e^{t}

Solution steps

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

Take the partial fraction of

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

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

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

L^{- 1} {\frac{1}{2 ( s + 3 )}} : \frac{1}{2} e^{- 3 t}

L^{- 1} {\frac{3}{2 ( s - 1 )}} : \frac{3}{2} e^{t}

= - \frac{1}{2} e^{- 3 t} + \frac{3}{2} e^{t}

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