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Beliebt Rechnen >

inverselaplace s/(s^2-s+1)

Lösung

inverse laplace transformation

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

Lösung

e^{\frac{t}{2}} cos (\frac{3 t}{2}) + \frac{e ^{\frac{t}{2}} sin ( \frac{3 t}{2} )}{3}

Schritte zur Lösung

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

Schreibe

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

um:

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

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

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})

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

Fasse

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

zusammen:

e^{\frac{t}{2}} cos (\frac{3 t}{2}) + \frac{e ^{\frac{t}{2}} sin ( \frac{3 t}{2} )}{3}

= e^{\frac{t}{2}} cos (\frac{3 t}{2}) + \frac{e ^{\frac{t}{2}} sin ( \frac{3 t}{2} )}{3}

Beliebte Beispiele

inverselaplace 3/(s(s+3))

in v erse l a pl a ce \frac{3}{s ( s + 3 )}

t an g e n t x^{2} + 3 x

(\partial)/(\partial x)(x^2y^3+xy-2)

\frac{\partial}{\partial x} (x^{2} y^{3} + x y - 2)

integral from ln(4) to ln(9) of e^{x/2}

\int_{l n (4)}^{l n (9)} e^{\frac{x}{2}} d x

derivative f(x)=sin(1/(x^2))

d er i v a t i v e f (x) = sin (\frac{1}{x ^{2}})