laplacetransform (1+t)^7

解答

拉普拉斯变换

(1 + t)^{7}

\frac{1}{s} + \frac{7}{s ^{2}} + \frac{42}{s ^{3}} + \frac{210}{s ^{4}} + \frac{840}{s ^{5}} + \frac{2520}{s ^{6}} + \frac{5040}{s ^{7}} + \frac{5040}{s ^{8}}

求解步骤

L {(1 + t)^{7}}

展开

(1 + t)^{7} : 1 + 7 t + 21 t^{2} + 35 t^{3} + 35 t^{4} + 21 t^{5} + 7 t^{6} + t^{7}

= L {1 + 7 t + 21 t^{2} + 35 t^{3} + 35 t^{4} + 21 t^{5} + 7 t^{6} + t^{7}}

利用拉普拉斯变换的线性特性:

对于函数

f (t), g (t)

和常数

a, b : L {a \cdot f (t) + b \cdot g (t)} = a \cdot L {f (t)} + b \cdot L {g (t)}

= L {1} + 7 L {t} + 21 L {t^{2}} + 35 L {t^{3}} + 35 L {t^{4}} + 21 L {t^{5}} + 7 L {t^{6}} + L {t^{7}}

L {1} : \frac{1}{s}

L {t} : \frac{1}{s ^{2}}

L {t^{2}} : \frac{2}{s ^{3}}

L {t^{3}} : \frac{6}{s ^{4}}

L {t^{4}} : \frac{24}{s ^{5}}

L {t^{5}} : \frac{120}{s ^{6}}

L {t^{6}} : \frac{720}{s ^{7}}

L {t^{7}} : \frac{5040}{s ^{8}}

= \frac{1}{s} + 7 \cdot \frac{1}{s ^{2}} + 21 \cdot \frac{2}{s ^{3}} + 35 \cdot \frac{6}{s ^{4}} + 35 \cdot \frac{24}{s ^{5}} + 21 \cdot \frac{120}{s ^{6}} + 7 \cdot \frac{720}{s ^{7}} + \frac{5040}{s ^{8}}

整理

\frac{1}{s} + 7 \frac{1}{s ^{2}} + 21 \frac{2}{s ^{3}} + 35 \frac{6}{s ^{4}} + 35 \frac{24}{s ^{5}} + 21 \frac{120}{s ^{6}} + 7 \frac{720}{s ^{7}} + \frac{5040}{s ^{8}} : \frac{1}{s} + \frac{7}{s ^{2}} + \frac{42}{s ^{3}} + \frac{210}{s ^{4}} + \frac{840}{s ^{5}} + \frac{2520}{s ^{6}} + \frac{5040}{s ^{7}} + \frac{5040}{s ^{8}}

= \frac{1}{s} + \frac{7}{s ^{2}} + \frac{42}{s ^{3}} + \frac{210}{s ^{4}} + \frac{840}{s ^{5}} + \frac{2520}{s ^{6}} + \frac{5040}{s ^{7}} + \frac{5040}{s ^{8}}

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