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Beliebt Rechnen >

inverselaplace 1/s*1/((s^2+3s+4))

Lösung

inverse laplace transformation

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

Lösung

\frac{1}{4} H (t) - \frac{1}{4} e^{- \frac{3 t}{2}} cos (\frac{7 t}{2}) - \frac{3 e ^{- \frac{3 t}{2}} sin ( \frac{7 t}{2} )}{4 7}

Schritte zur Lösung

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

Schreibe

\frac{1}{s} \frac{1}{s ^{2} + 3 s + 4}

um:

\frac{1}{s ^{3} + 3 s ^{2} + 4 s}

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

Ermittle den Partialbruch von

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

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

Schreibe um

= L^{- 1} {\frac{1}{4 s} + \frac{- s - 3}{4 s ^{2} + 12 s + 16}}

Schreibe

\frac{- s - 3}{4 s ^{2} + 12 s + 16}

um:

- \frac{1}{4} \cdot \frac{s + \frac{3}{2}}{( s + \frac{3}{2} ) ^{2} + \frac{7}{4}} - \frac{3}{8} \cdot \frac{1}{( s + \frac{3}{2} ) ^{2} + \frac{7}{4}}

= L^{- 1} {\frac{1}{4 s} - \frac{1}{4} \cdot \frac{s + \frac{3}{2}}{( s + \frac{3}{2} ) ^{2} + \frac{7}{4}} - \frac{3}{8} \cdot \frac{1}{( s + \frac{3}{2} ) ^{2} + \frac{7}{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{1}{4 s}} - \frac{1}{4} L^{- 1} {\frac{s + \frac{3}{2}}{( s + \frac{3}{2} ) ^{2} + \frac{7}{4}}} - \frac{3}{8} L^{- 1} {\frac{1}{( s + \frac{3}{2} ) ^{2} + \frac{7}{4}}}

L^{- 1} {\frac{1}{4 s}} : \frac{1}{4} H (t)

L^{- 1} {\frac{s + \frac{3}{2}}{( s + \frac{3}{2} ) ^{2} + \frac{7}{4}}} : e^{- \frac{3 t}{2}} cos (\frac{7 t}{2})

L^{- 1} {\frac{1}{( s + \frac{3}{2} ) ^{2} + \frac{7}{4}}} : e^{- \frac{3 t}{2}} \frac{2}{7} sin (\frac{7 t}{2})

= \frac{1}{4} H (t) - \frac{1}{4} e^{- \frac{3 t}{2}} cos (\frac{7 t}{2}) - \frac{3}{8} e^{- \frac{3 t}{2}} \frac{2}{7} sin (\frac{7 t}{2})

Fasse

\frac{1}{4} H (t) - \frac{1}{4} e^{- \frac{3 t}{2}} cos (\frac{7 t}{2}) - \frac{3}{8} e^{- \frac{3 t}{2}} \frac{2}{7} sin (\frac{7 t}{2})

zusammen:

\frac{1}{4} H (t) - \frac{1}{4} e^{- \frac{3 t}{2}} cos (\frac{7 t}{2}) - \frac{3 e ^{- \frac{3 t}{2}} sin ( \frac{7 t}{2} )}{4 7}

= \frac{1}{4} H (t) - \frac{1}{4} e^{- \frac{3 t}{2}} cos (\frac{7 t}{2}) - \frac{3 e ^{- \frac{3 t}{2}} sin ( \frac{7 t}{2} )}{4 7}

Beliebte Beispiele

integral of sin^3(xco)s^4x

\int sin^{3} (x co) s^{4} x d x

derivative of 2+4x^{3/2}

\frac{d}{d x} (2 + 4 x^{\frac{3}{2}})

derivative of ln(1-5x^2+x^3)

\frac{d}{d x} (ln (1 - 5 x^{2} + x^{3}))

derivative x^4+9e^x

d er i v a t i v e x^{4} + 9 e^{x}

tangent f(x)=x^2-3

t an g e n t f (x) = x^{2} - 3