inverselaplace 4/(s^4+4)

Lösung

inverse laplace transformation

\frac{4}{s ^{4} + 4}

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

Schritte zur Lösung

L^{- 1} {\frac{4}{s ^{4} + 4}}

Ermittle den Partialbruch von

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

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

Schreibe

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

um:

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

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

Schreibe

\frac{- s + 2}{2 ( s ^{2} - 2 s + 2 )}

um:

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

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

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

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

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

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

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

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

\int (2 x + 3)^{10} d x

y^{^{''}} + 9 y^{^{'}} + 20 y = 0

a re a 4 + x, 4 + \frac{1}{5} x

d er i v a t i v e f (x) = 2 x^{4}

s l o p e in t erce pt (- 3, - 5), (- 2, - 3)