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Beliebt Rechnen >

inverselaplace (10)/(s^2+s+10)*1/s

Lösung

inverse laplace transformation

\frac{10}{s ^{2} + s + 10} \cdot \frac{1}{s}

Lösung

H (t) - e^{- \frac{t}{2}} cos (\frac{39 t}{2}) - \frac{e ^{- \frac{t}{2}} sin ( \frac{39 t}{2} )}{39}

Schritte zur Lösung

L^{- 1} {\frac{10}{s ^{2} + s + 10} \cdot \frac{1}{s}}

Schreibe

\frac{10}{s ^{2} + s + 10} \frac{1}{s}

um:

\frac{10}{s ^{3} + s ^{2} + 10 s}

= L^{- 1} {\frac{10}{s ^{3} + s ^{2} + 10 s}}

Ermittle den Partialbruch von

\frac{10}{s ^{3} + s ^{2} + 10 s} : \frac{1}{s} + \frac{- s - 1}{s ^{2} + s + 10}

= L^{- 1} {\frac{1}{s} + \frac{- s - 1}{s ^{2} + s + 10}}

Schreibe

\frac{- s - 1}{s ^{2} + s + 10}

um:

- \frac{s + \frac{1}{2}}{( s + \frac{1}{2} ) ^{2} + \frac{39}{4}} - \frac{1}{2} \cdot \frac{1}{( s + \frac{1}{2} ) ^{2} + \frac{39}{4}}

= L^{- 1} {\frac{1}{s} - \frac{s + \frac{1}{2}}{( s + \frac{1}{2} ) ^{2} + \frac{39}{4}} - \frac{1}{2} \cdot \frac{1}{( s + \frac{1}{2} ) ^{2} + \frac{39}{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)}

= L^{- 1} {\frac{1}{s}} - L^{- 1} {\frac{s + \frac{1}{2}}{( s + \frac{1}{2} ) ^{2} + \frac{39}{4}}} - \frac{1}{2} L^{- 1} {\frac{1}{( s + \frac{1}{2} ) ^{2} + \frac{39}{4}}}

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

L^{- 1} {\frac{s + \frac{1}{2}}{( s + \frac{1}{2} ) ^{2} + \frac{39}{4}}} : e^{- \frac{t}{2}} cos (\frac{39 t}{2})

L^{- 1} {\frac{1}{( s + \frac{1}{2} ) ^{2} + \frac{39}{4}}} : e^{- \frac{t}{2}} \frac{2}{39} sin (\frac{39 t}{2})

= H (t) - e^{- \frac{t}{2}} cos (\frac{39 t}{2}) - \frac{1}{2} e^{- \frac{t}{2}} \frac{2}{39} sin (\frac{39 t}{2})

Fasse

H (t) - e^{- \frac{t}{2}} cos (\frac{39 t}{2}) - \frac{1}{2} e^{- \frac{t}{2}} \frac{2}{39} sin (\frac{39 t}{2})

zusammen:

H (t) - e^{- \frac{t}{2}} cos (\frac{39 t}{2}) - \frac{e ^{- \frac{t}{2}} sin ( \frac{39 t}{2} )}{39}

= H (t) - e^{- \frac{t}{2}} cos (\frac{39 t}{2}) - \frac{e ^{- \frac{t}{2}} sin ( \frac{39 t}{2} )}{39}

Beliebte Beispiele

(\partial)/(\partial x)((e^y)/(x+y^5))

\frac{\partial}{\partial x} (\frac{e ^{y}}{x + y ^{5}})

(\partial)/(\partial x)(cos(2cx+act))

\frac{\partial}{\partial x} (cos (2 c x + a c t))

derivative of 1000(1-x/(60)^2)

\frac{d}{d x} (1000 (1 - \frac{x}{60})^{2})

y^{^{'}} = 1 + \frac{y}{x}

y^{^{''}} - y = - x