de

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Beliebt Rechnen >

inverselaplace s/(s^4+1)

Lösung

inverse laplace transformation

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

Lösung

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

Schritte zur Lösung

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

Ermittle den Partialbruch von

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

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

Schreibe

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

um:

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

= L^{- 1} ⎩ ⎨ ⎧ - \frac{1}{2 2} \cdot \frac{1}{( s + \frac{1}{2} ) ^{2} + \frac{1}{2}} + \frac{1}{2 2 ( s ^{2} - 2 s + 1 )} ⎭ ⎬ ⎫

Schreibe

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

um:

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

= L^{- 1} ⎩ ⎨ ⎧ - \frac{1}{2 2} \cdot \frac{1}{( s + \frac{1}{2} ) ^{2} + \frac{1}{2}} + \frac{1}{2 2} \cdot \frac{1}{( s - \frac{1}{2} ) ^{2} + \frac{1}{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}{2 2} L^{- 1} ⎩ ⎨ ⎧ \frac{1}{( s + \frac{1}{2} ) ^{2} + \frac{1}{2}} ⎭ ⎬ ⎫ + \frac{1}{2 2} L^{- 1} ⎩ ⎨ ⎧ \frac{1}{( s - \frac{1}{2} ) ^{2} + \frac{1}{2}} ⎭ ⎬ ⎫

L^{- 1} ⎩ ⎨ ⎧ \frac{1}{( s + \frac{1}{2} ) ^{2} + \frac{1}{2}} ⎭ ⎬ ⎫ : e^{- \frac{t}{2}} 2 sin (\frac{1}{2} t)

L^{- 1} ⎩ ⎨ ⎧ \frac{1}{( s - \frac{1}{2} ) ^{2} + \frac{1}{2}} ⎭ ⎬ ⎫ : e^{\frac{t}{2}} 2 sin (\frac{1}{2} t)

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

Fasse

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

zusammen:

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

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

Beliebte Beispiele

d er i v a t i v e \frac{y}{x}

((100)/(log_{10)(2x+1)})^'

(\frac{100}{lo g _{10} ( 2 x + 1 )})^{^{'}}

f (x) = x^{3} e^{2 x}

limit as x approaches 1 of x^{3/(1-x)}

x \to 1 lim (x^{\frac{3}{1 - x}})

y^{''}-5y^'-6y=0,y(0)=5,y^'(0)=12

y^{^{''}} - 5 y^{^{'}} - 6 y = 0, y (0) = 5, y^{^{'}} (0) = 12