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

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

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inverse oflaplace (s+2)/(s(s+2)+1)

inverse laplace

\frac{s + 2}{s ( s + 2 ) + 1}

e^{- t} + e^{- t} t

Solution steps

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

Take the partial fraction of

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

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

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

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

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

= e^{- t} + e^{- t} t

\frac{d}{d x} (\frac{3 x + 1}{2 x - 1})

\frac{1}{s ( s + 4 )}

(6 e^{x} cos (x)) (0.6)

sin (\frac{1}{2} x)

\frac{\partial}{\partial x} (\frac{12 x}{6 x ^{2} + y ^{2} + 4})

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