inverselaplace 2/(s^2(s+2))

Lösung

inverse laplace transformation

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

- \frac{1}{2} H (t) + t + \frac{1}{2} e^{- 2 t}

Schritte zur Lösung

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

Ermittle den Partialbruch von

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

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

Wende die lineare Eigenschaftder inversen Laplace-Transformation an:

Für Funktionen

f (s), g (s)

und Konstanten

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

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

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

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

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

\int \frac{16 x}{8 x ^{2} + 1} d x

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