inverselaplace ((2s+1))/(s(2s+10))

解答

的拉普拉斯逆变换

\frac{( 2 s + 1 )}{s ( 2 s + 10 )}

\frac{1}{10} H (t) + \frac{9}{10} e^{- 5 t}

求解步骤

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

将

\frac{2 s + 1}{s ( 2 s + 10 )}

用部份分式展开:

\frac{1}{10 s} + \frac{9}{10 ( s + 5 )}

= L^{- 1} {\frac{1}{10 s} + \frac{9}{10 ( s + 5 )}}

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}{10 s}} + L^{- 1} {\frac{9}{10 ( s + 5 )}}

L^{- 1} {\frac{1}{10 s}} : \frac{1}{10} H (t)

L^{- 1} {\frac{9}{10 ( s + 5 )}} : \frac{9}{10} e^{- 5 t}

= \frac{1}{10} H (t) + \frac{9}{10} e^{- 5 t}

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