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受欢迎的微积分 >

inverselaplace 4/((s+10)(s^2+16)^2)

解答

的拉普拉斯逆变换

\frac{4}{( s + 10 ) ( s ^{2} + 16 ) ^{2}}

解答

\frac{- 232 t sin ( 4 t ) - 580 t cos ( 4 t ) + 16 e ^{- 10 t} - 16 cos ( 4 t ) + 185 sin ( 4 t )}{53824}

求解步骤

L^{- 1} {\frac{4}{( s + 10 ) ( s ^{2} + 16 ) ^{2}}}

将

\frac{4}{( s + 10 ) ( s ^{2} + 16 ) ^{2}}

用部份分式展开:

\frac{1}{3364 ( s + 10 )} + \frac{- s + 10}{3364 ( s ^{2} + 16 )} + \frac{- s + 10}{29 ( s ^{2} + 16 ) ^{2}}

= L^{- 1} {\frac{1}{3364 ( s + 10 )} + \frac{- s + 10}{3364 ( s ^{2} + 16 )} + \frac{- s + 10}{29 ( s ^{2} + 16 ) ^{2}}}

乘开

= L^{- 1} {- \frac{s}{3364 ( s ^{2} + 16 )} + \frac{1}{3364 ( s + 10 )} + \frac{10}{3364 ( s ^{2} + 16 )} + \frac{10}{29 ( s ^{2} + 16 ) ^{2}} - \frac{s}{29 ( s ^{2} + 16 ) ^{2}}}

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)}

= \frac{1}{3364} L^{- 1} {\frac{1}{s + 10}} - \frac{1}{3364} L^{- 1} {\frac{s}{s ^{2} + 16}} + \frac{10}{3364} L^{- 1} {\frac{1}{s ^{2} + 16}} - \frac{1}{29} L^{- 1} {\frac{s}{( s ^{2} + 16 ) ^{2}}} + \frac{10}{29} L^{- 1} {\frac{1}{( s ^{2} + 16 ) ^{2}}}

L^{- 1} {\frac{1}{s + 10}} : e^{- 10 t}

L^{- 1} {\frac{s}{s ^{2} + 16}} : cos (4 t)

L^{- 1} {\frac{1}{s ^{2} + 16}} : \frac{1}{4} sin (4 t)

L^{- 1} {\frac{s}{( s ^{2} + 16 ) ^{2}}} : \frac{t sin ( 4 t )}{8}

L^{- 1} {\frac{1}{( s ^{2} + 16 ) ^{2}}} : \frac{1}{128} (sin (4 t) - 4 t cos (4 t))

= \frac{1}{3364} e^{- 10 t} - \frac{1}{3364} cos (4 t) + \frac{10}{3364} \cdot \frac{1}{4} sin (4 t) - \frac{1}{29} \cdot \frac{t sin ( 4 t )}{8} + \frac{10}{29} \cdot \frac{1}{128} (sin (4 t) - 4 t cos (4 t))

化简

\frac{1}{3364} e^{- 10 t} - \frac{1}{3364} cos (4 t) + \frac{10}{3364} \frac{1}{4} sin (4 t) - \frac{1}{29} \frac{t sin ( 4 t )}{8} + \frac{10}{29} \frac{1}{128} (sin (4 t) - 4 t cos (4 t)) : \frac{- 232 t sin ( 4 t ) - 580 t cos ( 4 t ) + 16 e ^{- 10 t} - 16 cos ( 4 t ) + 185 sin ( 4 t )}{53824}

= \frac{- 232 t sin ( 4 t ) - 580 t cos ( 4 t ) + 16 e ^{- 10 t} - 16 cos ( 4 t ) + 185 sin ( 4 t )}{53824}

流行的例子

derivative y=(x^2+2x-4)/((x+1)^2)

d er i v a t i v e y = \frac{x ^{2} + 2 x - 4}{( x + 1 ) ^{2}}

integral of (6x+5)/(x^2+3x-4)

\int \frac{6 x + 5}{x ^{2} + 3 x - 4} d x

integral of (4x+e^x)sin(6x^2+3e^x)

\int (4 x + e^{x}) sin (6 x^{2} + 3 e^{x}) d x

integral of (1-x)/(1+x^2)

\int \frac{1 - x}{1 + x ^{2}} d x

derivative of (x^2+1^{x+1})

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