inverselaplace (0.5)/(s*(s^2+s+1))

Lösung

inverse laplace transformation

\frac{0.5}{s \cdot ( s ^{2} + s + 1 )}

0.5 H (t) - 0.5 e^{- \frac{t}{2}} cos (0.86602 \dots t) - 0.28867 \dots e^{- \frac{t}{2}} sin (0.86602 \dots t)

Schritte zur Lösung

L^{- 1} {\frac{0.5}{s ( s ^{2} + s + 1 )}}

Ermittle den Partialbruch von

\frac{0.5}{s ( s ^{2} + s + 1 )} : \frac{0.5}{s} + \frac{- 0.5 s - 0.5}{s ^{2} + s + 1}

= L^{- 1} {\frac{0.5}{s} + \frac{- 0.5 s - 0.5}{s ^{2} + s + 1}}

Schreibe

\frac{- 0.5 s - 0.5}{s ^{2} + s + 1}

um:

- 0.5 \cdot \frac{s + \frac{1}{2}}{( s + \frac{1}{2} ) ^{2} + \frac{3}{4}} - 0.25 \cdot \frac{1}{( s + \frac{1}{2} ) ^{2} + \frac{3}{4}}

= L^{- 1} {\frac{0.5}{s} - 0.5 \cdot \frac{s + \frac{1}{2}}{( s + \frac{1}{2} ) ^{2} + \frac{3}{4}} - 0.25 \cdot \frac{1}{( s + \frac{1}{2} ) ^{2} + \frac{3}{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{0.5}{s}} - 0.5 L^{- 1} {\frac{s + \frac{1}{2}}{( s + \frac{1}{2} ) ^{2} + \frac{3}{4}}} - 0.25 L^{- 1} {\frac{1}{( s + \frac{1}{2} ) ^{2} + \frac{3}{4}}}

L^{- 1} {\frac{0.5}{s}} : 0.5 H (t)

L^{- 1} {\frac{s + \frac{1}{2}}{( s + \frac{1}{2} ) ^{2} + \frac{3}{4}}} : e^{- \frac{t}{2}} cos (\frac{3 t}{2})

L^{- 1} {\frac{1}{( s + \frac{1}{2} ) ^{2} + \frac{3}{4}}} : e^{- \frac{t}{2}} \frac{2}{3} sin (\frac{3 t}{2})

= 0.5 H (t) - 0.5 e^{- \frac{t}{2}} cos (\frac{3 t}{2}) - 0.25 e^{- \frac{t}{2}} \frac{2}{3} sin (\frac{3 t}{2})

Vereinfache

0.5 H (t) - 0.5 e^{- \frac{t}{2}} cos (\frac{3 t}{2}) - 0.25 e^{- \frac{t}{2}} \frac{2}{3} sin (\frac{3 t}{2}) : 0.5 H (t) - 0.5 e^{- \frac{t}{2}} cos (0.86602 \dots t) - 0.28867 \dots e^{- \frac{t}{2}} sin (0.86602 \dots t)

= 0.5 H (t) - 0.5 e^{- \frac{t}{2}} cos (0.86602 \dots t) - 0.28867 \dots e^{- \frac{t}{2}} sin (0.86602 \dots t)

\frac{d x}{d t} - (tan (t)) x = sin (t), x (0) = 1

x \to 0 lim (\frac{x}{x ^{2} + 7 x})

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

\int_{- 3}^{5} 6 - y - \frac{( y - 3 ) ^{2}}{4} d y

d er i v a t i v e f (x) = 7 tan^{3} (x)