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受欢迎的微积分 >

inverselaplace 1/(s^3+1)

解答

的拉普拉斯逆变换

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

解答

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

求解步骤

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

将

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

用部份分式展开:

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

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

展开

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

利用逆拉普拉斯变换的线性特性:

对于函数

f (s), g (s)

和常数

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

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

整理

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

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

流行的例子

limit as x approaches 0 of x*e^{-1/x}

x \to 0 lim (x \cdot e^{- \frac{1}{x}})

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\int \frac{x ^{2}}{( x + 1 ) ^{4}} d x

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x \to 0 lim (x - \frac{4}{x ^{2}})

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\frac{d}{d x} (arcsech (sin (x)))

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\int e^{x} x^{4} d x