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

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

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

inverse laplace

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

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

Solution steps

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

Take the partial fraction of

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

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

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

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

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

y^{^{'}} + y cos (x) = 4 cos (x)

\frac{d}{d x} (\frac{1 - cos ( x )}{x ^{2}})

\int sin (2 x) 1 + 2 cos (2 x) d x

x \to 2 lim (2 (3))

\int cos^{4} (5 x + 7) d x

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