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Beliebt Rechnen >

laplacetransform t^2e^{-2t}sin(3t+30)

Lösung

laplace transformation

t^{2} e^{- 2 t} sin (3 t + 30)

Lösung

- \frac{2 ( - sin ( 30 ) s ^{3} - 6 sin ( 30 ) s ^{2} - 9 cos ( 30 ) s ^{2} + 15 sin ( 30 ) s - 36 cos ( 30 ) s + 46 sin ( 30 ) - 9 cos ( 30 ) )}{( s ^{2} + 4 s + 13 ) ^{3}}

Schritte zur Lösung

L {e^{- 2 t} t^{2} sin (3 t + 30)}

Wende Laplace-Transformationstabelle an:

L {t^{k} f (t)} = (- 1)^{k} \frac{d ^{k}}{d s ^{k}} (L {f (t)})

Für

t^{2} e^{- 2 t} sin (3 t + 30) : f (t) = e^{- 2 t} sin (3 t + 30), k = 2

= (- 1)^{2} \frac{d ^{2}}{d s ^{2}} (L {e^{- 2 t} sin (3 t + 30)})

L {e^{- 2 t} sin (3 t + 30)} : \frac{sin ( 30 ) ( s + 2 ) + 3 cos ( 30 )}{( s + 2 ) ^{2} + 9}

\frac{d ^{2}}{d s ^{2}} (\frac{sin ( 30 ) ( s + 2 ) + 3 cos ( 30 )}{( s + 2 ) ^{2} + 9}) = - \frac{2 ( - sin ( 30 ) s ^{3} - 6 sin ( 30 ) s ^{2} - 9 cos ( 30 ) s ^{2} + 15 sin ( 30 ) s - 36 cos ( 30 ) s + 46 sin ( 30 ) - 9 cos ( 30 ) )}{( s ^{2} + 4 s + 13 ) ^{3}}

= (- 1)^{2} (- \frac{2 ( - sin ( 30 ) s ^{3} - 6 sin ( 30 ) s ^{2} - 9 cos ( 30 ) s ^{2} + 15 sin ( 30 ) s - 36 cos ( 30 ) s + 46 sin ( 30 ) - 9 cos ( 30 ) )}{( s ^{2} + 4 s + 13 ) ^{3}})

= - \frac{2 ( - sin ( 30 ) s ^{3} - 6 sin ( 30 ) s ^{2} - 9 cos ( 30 ) s ^{2} + 15 sin ( 30 ) s - 36 cos ( 30 ) s + 46 sin ( 30 ) - 9 cos ( 30 ) )}{( s ^{2} + 4 s + 13 ) ^{3}}

Beliebte Beispiele

(\partial)/(\partial x)(ysqrt(x^2+y^2))

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

limit as x approaches infinity of 3x^3-1000x^2

x \to \infty lim (3 x^{3} - 1000 x^{2})

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\int 121 - t^{2} d t

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\frac{d}{d x} (7 x y)