What is inverse oflaplace S/(S^2+2S-3) ?

The answer to inverse oflaplace S/(S^2+2S-3) is 1/4 e^t+3/4 e^{-3t}

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inverse oflaplace S/(S^2+2S-3)

inverse laplace

\frac{S}{S ^{2} + 2 S - 3}

\frac{1}{4} e^{t} + \frac{3}{4} e^{- 3 t}

Solution steps

L^{- 1} {\frac{S}{S ^{2} + 2 S - 3}}

Take the partial fraction of

\frac{S}{S ^{2} + 2 S - 3} : \frac{1}{4 ( S - 1 )} + \frac{3}{4 ( S + 3 )}

= L^{- 1} {\frac{1}{4 ( S - 1 )} + \frac{3}{4 ( 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} {\frac{1}{4 ( S - 1 )}} + L^{- 1} {\frac{3}{4 ( S + 3 )}}

L^{- 1} {\frac{1}{4 ( S - 1 )}} : \frac{1}{4} e^{t}

L^{- 1} {\frac{3}{4 ( S + 3 )}} : \frac{3}{4} e^{- 3 t}

= \frac{1}{4} e^{t} + \frac{3}{4} e^{- 3 t}

What is inverse oflaplace S/(S^2+2S-3) ?
The answer to inverse oflaplace S/(S^2+2S-3) is 1/4 e^t+3/4 e^{-3t}