inverselaplace 8/(s^2(s-2)^3)

Lösung

inverse laplace transformation

\frac{8}{s ^{2} ( s - 2 ) ^{3}}

- \frac{3}{2} H (t) - t + \frac{3}{2} e^{2 t} - e^{2 t} \cdot 2 t + e^{2 t} t^{2}

Schritte zur Lösung

L^{- 1} {\frac{8}{s ^{2} ( s - 2 ) ^{3}}}

Ermittle den Partialbruch von

\frac{8}{s ^{2} ( s - 2 ) ^{3}} : - \frac{3}{2 s} - \frac{1}{s ^{2}} + \frac{3}{2 ( s - 2 )} - \frac{2}{( s - 2 ) ^{2}} + \frac{2}{( s - 2 ) ^{3}}

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

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

L^{- 1} {\frac{3}{2 s}} : \frac{3}{2} H (t)

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

L^{- 1} {\frac{3}{2 ( s - 2 )}} : \frac{3}{2} e^{2 t}

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

L^{- 1} {\frac{2}{( s - 2 ) ^{3}}} : e^{2 t} t^{2}

= - \frac{3}{2} H (t) - t + \frac{3}{2} e^{2 t} - e^{2 t} \cdot 2 t + e^{2 t} t^{2}

\int x x - 5 d x

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a re a 7 x^{2} - 3 x, x^{2} - 7 x + 2

x \to + 0 + lim (\frac{e ^{x}}{e ^{x} - 1})

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