laplacetransform f(t)=t^2e^{3t}

解答

拉普拉斯变换

f (t) = t^{2} e^{3 t}

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

求解步骤

L {e^{3 t} t^{2}}

使用拉普拉斯变换表:

L {t^{k} f (t)} = (- 1)^{k} \frac{d ^{k}}{d s ^{k}} (L {f (t)})

对于

t^{2} e^{3 t} : f (t) = e^{3 t}, k = 2

= (- 1)^{2} \frac{d ^{2}}{d s ^{2}} (L {e^{3 t}})

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

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

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

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

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