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Beliebt Rechnen >

inverselaplace 4/((s+10)(s^2+16)^2)

Lösung

inverse laplace transformation

\frac{4}{( s + 10 ) ( s ^{2} + 16 ) ^{2}}

Lösung

\frac{- 232 t sin ( 4 t ) - 580 t cos ( 4 t ) + 16 e ^{- 10 t} - 16 cos ( 4 t ) + 185 sin ( 4 t )}{53824}

Schritte zur Lösung

L^{- 1} {\frac{4}{( s + 10 ) ( s ^{2} + 16 ) ^{2}}}

Ermittle den Partialbruch von

\frac{4}{( s + 10 ) ( s ^{2} + 16 ) ^{2}} : \frac{1}{3364 ( s + 10 )} + \frac{- s + 10}{3364 ( s ^{2} + 16 )} + \frac{- s + 10}{29 ( s ^{2} + 16 ) ^{2}}

= L^{- 1} {\frac{1}{3364 ( s + 10 )} + \frac{- s + 10}{3364 ( s ^{2} + 16 )} + \frac{- s + 10}{29 ( s ^{2} + 16 ) ^{2}}}

Schreibe um

= L^{- 1} {- \frac{s}{3364 ( s ^{2} + 16 )} + \frac{1}{3364 ( s + 10 )} + \frac{10}{3364 ( s ^{2} + 16 )} + \frac{10}{29 ( s ^{2} + 16 ) ^{2}} - \frac{s}{29 ( s ^{2} + 16 ) ^{2}}}

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{1}{3364} L^{- 1} {\frac{1}{s + 10}} - \frac{1}{3364} L^{- 1} {\frac{s}{s ^{2} + 16}} + \frac{10}{3364} L^{- 1} {\frac{1}{s ^{2} + 16}} - \frac{1}{29} L^{- 1} {\frac{s}{( s ^{2} + 16 ) ^{2}}} + \frac{10}{29} L^{- 1} {\frac{1}{( s ^{2} + 16 ) ^{2}}}

L^{- 1} {\frac{1}{s + 10}} : e^{- 10 t}

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

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

L^{- 1} {\frac{s}{( s ^{2} + 16 ) ^{2}}} : \frac{t sin ( 4 t )}{8}

L^{- 1} {\frac{1}{( s ^{2} + 16 ) ^{2}}} : \frac{1}{128} (sin (4 t) - 4 t cos (4 t))

= \frac{1}{3364} e^{- 10 t} - \frac{1}{3364} cos (4 t) + \frac{10}{3364} \cdot \frac{1}{4} sin (4 t) - \frac{1}{29} \cdot \frac{t sin ( 4 t )}{8} + \frac{10}{29} \cdot \frac{1}{128} (sin (4 t) - 4 t cos (4 t))

Vereinfache

\frac{1}{3364} e^{- 10 t} - \frac{1}{3364} cos (4 t) + \frac{10}{3364} \frac{1}{4} sin (4 t) - \frac{1}{29} \frac{t sin ( 4 t )}{8} + \frac{10}{29} \frac{1}{128} (sin (4 t) - 4 t cos (4 t)) : \frac{- 232 t sin ( 4 t ) - 580 t cos ( 4 t ) + 16 e ^{- 10 t} - 16 cos ( 4 t ) + 185 sin ( 4 t )}{53824}

= \frac{- 232 t sin ( 4 t ) - 580 t cos ( 4 t ) + 16 e ^{- 10 t} - 16 cos ( 4 t ) + 185 sin ( 4 t )}{53824}

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\frac{d}{d x} ((x^{2} + 1)^{x + 1})