de

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Beliebt Rechnen >

inverselaplace (s+2)/(s^2+2s+5)*1/s

Lösung

inverse laplace transformation

\frac{s + 2}{s ^{2} + 2 s + 5} \cdot \frac{1}{s}

Lösung

\frac{2}{5} H (t) - \frac{2}{5} e^{- t} cos (2 t) + \frac{3}{10} e^{- t} sin (2 t)

Schritte zur Lösung

L^{- 1} {\frac{1}{s} \cdot \frac{s + 2}{s ^{2} + 2 s + 5}}

Schreibe

\frac{s + 2}{s ^{2} + 2 s + 5} \frac{1}{s}

um:

\frac{s + 2}{s ^{3} + 2 s ^{2} + 5 s}

= L^{- 1} {\frac{s + 2}{s ^{3} + 2 s ^{2} + 5 s}}

Ermittle den Partialbruch von

\frac{s + 2}{s ^{3} + 2 s ^{2} + 5 s} : \frac{2}{5 s} + \frac{- 2 s + 1}{5 ( s ^{2} + 2 s + 5 )}

= L^{- 1} {\frac{2}{5 s} + \frac{- 2 s + 1}{5 ( s ^{2} + 2 s + 5 )}}

Schreibe um

= L^{- 1} {\frac{2}{5 s} + \frac{- 2 s + 1}{5 s ^{2} + 10 s + 25}}

Schreibe

\frac{- 2 s + 1}{5 s ^{2} + 10 s + 25}

um:

- \frac{2}{5} \cdot \frac{s + 1}{( s + 1 ) ^{2} + 4} + \frac{3}{5} \cdot \frac{1}{( s + 1 ) ^{2} + 4}

= L^{- 1} {\frac{2}{5 s} - \frac{2}{5} \cdot \frac{s + 1}{( s + 1 ) ^{2} + 4} + \frac{3}{5} \cdot \frac{1}{( s + 1 ) ^{2} + 4}}

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{2}{5 s}} - \frac{2}{5} L^{- 1} {\frac{s + 1}{( s + 1 ) ^{2} + 4}} + \frac{3}{5} L^{- 1} {\frac{1}{( s + 1 ) ^{2} + 4}}

L^{- 1} {\frac{2}{5 s}} : \frac{2}{5} H (t)

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

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

= \frac{2}{5} H (t) - \frac{2}{5} e^{- t} cos (2 t) + \frac{3}{5} e^{- t} \frac{1}{2} sin (2 t)

Fasse

\frac{2}{5} H (t) - \frac{2}{5} e^{- t} cos (2 t) + \frac{3}{5} e^{- t} \frac{1}{2} sin (2 t)

zusammen:

\frac{2}{5} H (t) - \frac{2}{5} e^{- t} cos (2 t) + \frac{3}{10} e^{- t} sin (2 t)

= \frac{2}{5} H (t) - \frac{2}{5} e^{- t} cos (2 t) + \frac{3}{10} e^{- t} sin (2 t)

Beliebte Beispiele

(d^2)/(dx^2)(2/x)

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

integral of ((sin(x))/x)^2

\int (\frac{sin ( x )}{x})^{2} d x

y^{''}-12y^'+37y=0,y(0)=6,y^'(0)=8

y^{^{''}} - 12 y^{^{'}} + 37 y = 0, y (0) = 6, y^{^{'}} (0) = 8

derivative of 3(5x^4+5x^3^4)

\frac{d}{d x} (3 (5 x^{4} + 5 x^{3})^{4})

integral of (x^2+12x+12)/(x^3-4x)

\int \frac{x ^{2} + 12 x + 12}{x ^{3} - 4 x} d x