laplacetransform (t-2)^3(t-2)

Lösung

laplace transformation

(t - 2)^{3} (t - 2)

\frac{24}{s ^{5}} - \frac{48}{s ^{4}} + \frac{48}{s ^{3}} - \frac{32}{s ^{2}} + \frac{16}{s}

Schritte zur Lösung

L {(t - 2)^{3} (t - 2)}

Schreibe

(t - 2)^{3} (t - 2)

um:

t^{4} - 8 t^{3} + 24 t^{2} - 32 t + 16

= L {t^{4} - 8 t^{3} + 24 t^{2} - 32 t + 16}

Verwende die lineare Eigenschaft der Laplace-Transformation:

Für Funktionen

f (t), g (t)

und Konstanten

a, b : L {a \cdot f (t) + b \cdot g (t)} = a \cdot L {f (t)} + b \cdot L {g (t)}

= L {t^{4}} - 8 L {t^{3}} + 24 L {t^{2}} - 32 L {t} + L {16}

L {t^{4}} : \frac{24}{s ^{5}}

L {t^{3}} : \frac{6}{s ^{4}}

L {t^{2}} : \frac{2}{s ^{3}}

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

L {16} : \frac{16}{s}

= \frac{24}{s ^{5}} - 8 \cdot \frac{6}{s ^{4}} + 24 \cdot \frac{2}{s ^{3}} - 32 \cdot \frac{1}{s ^{2}} + \frac{16}{s}

Fasse

\frac{24}{s ^{5}} - 8 \frac{6}{s ^{4}} + 24 \frac{2}{s ^{3}} - 32 \frac{1}{s ^{2}} + \frac{16}{s}

zusammen:

\frac{24}{s ^{5}} - \frac{48}{s ^{4}} + \frac{48}{s ^{3}} - \frac{32}{s ^{2}} + \frac{16}{s}

= \frac{24}{s ^{5}} - \frac{48}{s ^{4}} + \frac{48}{s ^{3}} - \frac{32}{s ^{2}} + \frac{16}{s}

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