inverselaplace ((2s+2))/((s^2+13s+12))

Lösung

inverse laplace transformation

\frac{( 2 s + 2 )}{( s ^{2} + 13 s + 12 )}

2 e^{- \frac{13 t}{2}} cosh (\frac{11 t}{2}) - 2 e^{- \frac{13 t}{2}} sinh (\frac{11 t}{2})

Schritte zur Lösung

L^{- 1} {\frac{( 2 s + 2 )}{( s ^{2} + 13 s + 12 )}}

Schreibe

\frac{2 s + 2}{s ^{2} + 13 s + 12}

um:

2 \cdot \frac{s + \frac{13}{2}}{( s + \frac{13}{2} ) ^{2} - \frac{121}{4}} - 11 \cdot \frac{1}{( s + \frac{13}{2} ) ^{2} - \frac{121}{4}}

= L^{- 1} {2 \cdot \frac{s + \frac{13}{2}}{( s + \frac{13}{2} ) ^{2} - \frac{121}{4}} - 11 \cdot \frac{1}{( s + \frac{13}{2} ) ^{2} - \frac{121}{4}}}

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)}

= 2 L^{- 1} {\frac{s + \frac{13}{2}}{( s + \frac{13}{2} ) ^{2} - \frac{121}{4}}} - 11 L^{- 1} {\frac{1}{( s + \frac{13}{2} ) ^{2} - \frac{121}{4}}}

L^{- 1} {\frac{s + \frac{13}{2}}{( s + \frac{13}{2} ) ^{2} - \frac{121}{4}}} : e^{- \frac{13 t}{2}} cosh (\frac{11 t}{2})

L^{- 1} {\frac{1}{( s + \frac{13}{2} ) ^{2} - \frac{121}{4}}} : e^{- \frac{13 t}{2}} \frac{2}{11} sinh (\frac{11 t}{2})

= 2 e^{- \frac{13 t}{2}} cosh (\frac{11 t}{2}) - 11 e^{- \frac{13 t}{2}} \frac{2}{11} sinh (\frac{11 t}{2})

Fasse

2 e^{- \frac{13 t}{2}} cosh (\frac{11 t}{2}) - 11 e^{- \frac{13 t}{2}} \frac{2}{11} sinh (\frac{11 t}{2})

zusammen:

2 e^{- \frac{13 t}{2}} cosh (\frac{11 t}{2}) - 2 e^{- \frac{13 t}{2}} sinh (\frac{11 t}{2})

= 2 e^{- \frac{13 t}{2}} cosh (\frac{11 t}{2}) - 2 e^{- \frac{13 t}{2}} sinh (\frac{11 t}{2})

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