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Beliebt Rechnen >

inverselaplace 1/((s+1)^2(s^2+4))

Lösung

inverse laplace transformation

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

Lösung

\frac{2}{25} e^{- t} + \frac{1}{5} e^{- t} t - \frac{2}{25} cos (2 t) - \frac{3}{50} sin (2 t)

Schritte zur Lösung

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

Ermittle den Partialbruch von

\frac{1}{( s + 1 ) ^{2} ( s ^{2} + 4 )} : \frac{2}{25 ( s + 1 )} + \frac{1}{5 ( s + 1 ) ^{2}} + \frac{- 2 s - 3}{25 ( s ^{2} + 4 )}

= L^{- 1} {\frac{2}{25 ( s + 1 )} + \frac{1}{5 ( s + 1 ) ^{2}} + \frac{- 2 s - 3}{25 ( s ^{2} + 4 )}}

Schreibe um

= L^{- 1} {- \frac{2 s}{25 ( s ^{2} + 4 )} + \frac{2}{25 ( s + 1 )} + \frac{1}{5 ( s + 1 ) ^{2}} - \frac{3}{25 ( s ^{2} + 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)}

= \frac{2}{25} L^{- 1} {\frac{1}{s + 1}} + \frac{1}{5} L^{- 1} {\frac{1}{( s + 1 ) ^{2}}} - \frac{2}{25} L^{- 1} {\frac{s}{s ^{2} + 4}} - \frac{3}{25} L^{- 1} {\frac{1}{s ^{2} + 4}}

L^{- 1} {\frac{1}{s + 1}} : e^{- t}

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

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

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

= \frac{2}{25} e^{- t} + \frac{1}{5} e^{- t} t - \frac{2}{25} cos (2 t) - \frac{3}{25} \cdot \frac{1}{2} sin (2 t)

Fasse

\frac{2}{25} e^{- t} + \frac{1}{5} e^{- t} t - \frac{2}{25} cos (2 t) - \frac{3}{25} \frac{1}{2} sin (2 t)

zusammen:

\frac{2}{25} e^{- t} + \frac{1}{5} e^{- t} t - \frac{2}{25} cos (2 t) - \frac{3}{50} sin (2 t)

= \frac{2}{25} e^{- t} + \frac{1}{5} e^{- t} t - \frac{2}{25} cos (2 t) - \frac{3}{50} sin (2 t)

Beliebte Beispiele

fläche y=sqrt(x+2),y= 1/(x+1),x=2

a re a y = x + 2, y = \frac{1}{x + 1}, x = 2

derivative of 2cot(x)

\frac{d}{d x} (2 cot (x))

(\partial)/(\partial x)(2x^2-2xy^2)

\frac{\partial}{\partial x} (2 x^{2} - 2 x y^{2})

integral of sqrt((x+3)/(x-1))

\int \frac{x + 3}{x - 1} d x

derivative of arcsin(x/pi)

\frac{d}{d x} (arcsin (\frac{x}{π}))