inverselaplace s/(s^2+23pi*s+pi^2)

Lösung

inverse laplace transformation

\frac{s}{s ^{2} + 2 \cdot 3 \cdot π \cdot s + π ^{2}}

e^{- 3 π t} cosh (22 π t) - \frac{3}{2 2} e^{- 3 π t} sinh (22 π t)

Schritte zur Lösung

L^{- 1} {\frac{s}{s ^{2} + 2 \cdot 3 π s + π ^{2}}}

Schreibe

\frac{s}{s ^{2} + 23 π s + π ^{2}}

um:

\frac{s + 3 π}{( s + 3 π ) ^{2} - 8 π ^{2}} - 3 π \frac{1}{( s + 3 π ) ^{2} - 8 π ^{2}}

= L^{- 1} {\frac{s + 3 π}{( s + 3 π ) ^{2} - 8 π ^{2}} - 3 π \frac{1}{( s + 3 π ) ^{2} - 8 π ^{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)}

= L^{- 1} {\frac{s + 3 π}{( s + 3 π ) ^{2} - 8 π ^{2}}} - 3 π L^{- 1} {\frac{1}{( s + 3 π ) ^{2} - 8 π ^{2}}}

L^{- 1} {\frac{s + 3 π}{( s + 3 π ) ^{2} - 8 π ^{2}}} : e^{- 3 π t} cosh (22 π t)

L^{- 1} {\frac{1}{( s + 3 π ) ^{2} - 8 π ^{2}}} : e^{- 3 π t} \frac{1}{2 2 π} sinh (22 π t)

= e^{- 3 π t} cosh (22 π t) - 3 π e^{- 3 π t} \frac{1}{2 2 π} sinh (22 π t)

Fasse

e^{- 3 π t} cosh (22 π t) - 3 π e^{- 3 π t} \frac{1}{2 2 π} sinh (22 π t)

zusammen:

e^{- 3 π t} cosh (22 π t) - \frac{3}{2 2} e^{- 3 π t} sinh (22 π t)

= e^{- 3 π t} cosh (22 π t) - \frac{3}{2 2} e^{- 3 π t} sinh (22 π t)

\frac{d}{d x} ((5 x + e^{x}) (5 - x))

\frac{d}{d t} (2 e^{2 t})

x \to 2 + lim (x - 2)

\frac{d}{d x} (ln (1 + \frac{a}{x}))

x \to + 3 lim (f (x))