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Beliebt Rechnen >

inverselaplace (s+3)/(s^2+5s+1)

Lösung

inverse laplace transformation

\frac{s + 3}{s ^{2} + 5 s + 1}

Lösung

e^{- \frac{5 t}{2}} cosh (\frac{21 t}{2}) + \frac{e ^{- \frac{5 t}{2}} sinh ( \frac{21 t}{2} )}{21}

Schritte zur Lösung

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

Schreibe

\frac{s + 3}{s ^{2} + 5 s + 1}

um:

\frac{s + \frac{5}{2}}{( s + \frac{5}{2} ) ^{2} - \frac{21}{4}} + \frac{1}{2} \cdot \frac{1}{( s + \frac{5}{2} ) ^{2} - \frac{21}{4}}

= L^{- 1} {\frac{s + \frac{5}{2}}{( s + \frac{5}{2} ) ^{2} - \frac{21}{4}} + \frac{1}{2} \cdot \frac{1}{( s + \frac{5}{2} ) ^{2} - \frac{21}{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{5}{2}}{( s + \frac{5}{2} ) ^{2} - \frac{21}{4}}} + \frac{1}{2} L^{- 1} {\frac{1}{( s + \frac{5}{2} ) ^{2} - \frac{21}{4}}}

L^{- 1} {\frac{s + \frac{5}{2}}{( s + \frac{5}{2} ) ^{2} - \frac{21}{4}}} : e^{- \frac{5 t}{2}} cosh (\frac{21 t}{2})

L^{- 1} {\frac{1}{( s + \frac{5}{2} ) ^{2} - \frac{21}{4}}} : e^{- \frac{5 t}{2}} \frac{2}{21} sinh (\frac{21 t}{2})

= e^{- \frac{5 t}{2}} cosh (\frac{21 t}{2}) + \frac{1}{2} e^{- \frac{5 t}{2}} \frac{2}{21} sinh (\frac{21 t}{2})

Fasse

e^{- \frac{5 t}{2}} cosh (\frac{21 t}{2}) + \frac{1}{2} e^{- \frac{5 t}{2}} \frac{2}{21} sinh (\frac{21 t}{2})

zusammen:

e^{- \frac{5 t}{2}} cosh (\frac{21 t}{2}) + \frac{e ^{- \frac{5 t}{2}} sinh ( \frac{21 t}{2} )}{21}

= e^{- \frac{5 t}{2}} cosh (\frac{21 t}{2}) + \frac{e ^{- \frac{5 t}{2}} sinh ( \frac{21 t}{2} )}{21}

Beliebte Beispiele

integral of (xe^{5x})

\int (x e^{5 x}) d x

(dy)/(e^{-2y)}=8x^3dx

\frac{d y}{e ^{- 2 y}} = 8 x^{3} d x

(\partial)/(\partial y)(9x^{y/z})

\frac{\partial}{\partial y} (9 x^{\frac{y}{z}})

derivative of (sin(3x)/(cos(2x)))

\frac{d}{d x} (\frac{sin ( 3 x )}{cos ( 2 x )})

derivative arcsin(x/9)

d er i v a t i v e arcsin (\frac{x}{9})