What is inverse oflaplace (s^2)/(s^2+4s+5) ?

The answer to inverse oflaplace (s^2)/(s^2+4s+5) is \delta(t)-4e^{-2t}cos(t)+3e^{-2t}sin(t)

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inverse oflaplace (s^2)/(s^2+4s+5)

inverse laplace

\frac{s ^{2}}{s ^{2} + 4 s + 5}

δ (t) - 4 e^{- 2 t} cos (t) + 3 e^{- 2 t} sin (t)

Solution steps

L^{- 1} {\frac{s ^{2}}{s ^{2} + 4 s + 5}}

Take the partial fraction of

\frac{s ^{2}}{s ^{2} + 4 s + 5} : 1 + \frac{- 4 s - 5}{s ^{2} + 4 s + 5}

= L^{- 1} {1 + \frac{- 4 s - 5}{s ^{2} + 4 s + 5}}

Expand

\frac{- 4 s - 5}{s ^{2} + 4 s + 5} : - 4 \cdot \frac{s + 2}{( s + 2 ) ^{2} + 1} + 3 \cdot \frac{1}{( s + 2 ) ^{2} + 1}

= L^{- 1} {1 - 4 \cdot \frac{s + 2}{( s + 2 ) ^{2} + 1} + 3 \cdot \frac{1}{( s + 2 ) ^{2} + 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} {1} - 4 L^{- 1} {\frac{s + 2}{( s + 2 ) ^{2} + 1}} + 3 L^{- 1} {\frac{1}{( s + 2 ) ^{2} + 1}}

L^{- 1} {1} : δ (t)

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

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

= δ (t) - 4 e^{- 2 t} cos (t) + 3 e^{- 2 t} sin (t)

What is inverse oflaplace (s^2)/(s^2+4s+5) ?
The answer to inverse oflaplace (s^2)/(s^2+4s+5) is \delta(t)-4e^{-2t}cos(t)+3e^{-2t}sin(t)