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Beliebt Rechnen >

inverselaplace 1/s*1/(s^2+s/2+1/2)

Lösung

inverse laplace transformation

\frac{1}{s} \cdot \frac{1}{s ^{2} + \frac{s}{2} + \frac{1}{2}}

Lösung

2 H (t) - 2 e^{- \frac{t}{4}} cos (\frac{7 t}{4}) - \frac{2 e ^{- \frac{t}{4}} sin ( \frac{7 t}{4} )}{7}

Schritte zur Lösung

L^{- 1} {\frac{1}{s} \cdot \frac{1}{s ^{2} + \frac{s}{2} + \frac{1}{2}}}

Schreibe

\frac{1}{s} \frac{1}{s ^{2} + \frac{s}{2} + \frac{1}{2}}

um:

\frac{2}{2 s ^{3} + s ^{2} + s}

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

Ermittle den Partialbruch von

\frac{2}{2 s ^{3} + s ^{2} + s} : \frac{2}{s} + \frac{- 4 s - 2}{2 s ^{2} + s + 1}

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

Schreibe

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

um:

- 2 \cdot \frac{s + \frac{1}{4}}{( s + \frac{1}{4} ) ^{2} + \frac{7}{16}} - \frac{1}{2} \cdot \frac{1}{( s + \frac{1}{4} ) ^{2} + \frac{7}{16}}

= L^{- 1} {\frac{2}{s} - 2 \cdot \frac{s + \frac{1}{4}}{( s + \frac{1}{4} ) ^{2} + \frac{7}{16}} - \frac{1}{2} \cdot \frac{1}{( s + \frac{1}{4} ) ^{2} + \frac{7}{16}}}

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{2}{s}} - 2 L^{- 1} {\frac{s + \frac{1}{4}}{( s + \frac{1}{4} ) ^{2} + \frac{7}{16}}} - \frac{1}{2} L^{- 1} {\frac{1}{( s + \frac{1}{4} ) ^{2} + \frac{7}{16}}}

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

L^{- 1} {\frac{s + \frac{1}{4}}{( s + \frac{1}{4} ) ^{2} + \frac{7}{16}}} : e^{- \frac{t}{4}} cos (\frac{7 t}{4})

L^{- 1} {\frac{1}{( s + \frac{1}{4} ) ^{2} + \frac{7}{16}}} : e^{- \frac{t}{4}} \frac{4}{7} sin (\frac{7 t}{4})

= 2 H (t) - 2 e^{- \frac{t}{4}} cos (\frac{7 t}{4}) - \frac{1}{2} e^{- \frac{t}{4}} \frac{4}{7} sin (\frac{7 t}{4})

Fasse

2 H (t) - 2 e^{- \frac{t}{4}} cos (\frac{7 t}{4}) - \frac{1}{2} e^{- \frac{t}{4}} \frac{4}{7} sin (\frac{7 t}{4})

zusammen:

2 H (t) - 2 e^{- \frac{t}{4}} cos (\frac{7 t}{4}) - \frac{2 e ^{- \frac{t}{4}} sin ( \frac{7 t}{4} )}{7}

= 2 H (t) - 2 e^{- \frac{t}{4}} cos (\frac{7 t}{4}) - \frac{2 e ^{- \frac{t}{4}} sin ( \frac{7 t}{4} )}{7}

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\int \frac{1}{x ^{2} + 8 x + 52} d x

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