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

解答

的拉普拉斯逆变换

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

\frac{1}{6} H (t) - \frac{1}{2} e^{- 2 t} + \frac{1}{3} e^{- 3 t}

求解步骤

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

将

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

用部份分式展开:

\frac{1}{6 s} - \frac{1}{2 ( s + 2 )} + \frac{1}{3 ( s + 3 )}

= L^{- 1} {\frac{1}{6 s} - \frac{1}{2 ( s + 2 )} + \frac{1}{3 ( s + 3 )}}

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

L^{- 1} {\frac{1}{6 s}} : \frac{1}{6} H (t)

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

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

= \frac{1}{6} H (t) - \frac{1}{2} e^{- 2 t} + \frac{1}{3} e^{- 3 t}

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