laplacetransform e^{2t}(t-1)^2

解答

拉普拉斯变换

e^{2 t} (t - 1)^{2}

\frac{s ^{2} - 6 s + 10}{( s - 2 ) ^{3}}

求解步骤

L {e^{2 t} (t - 1)^{2}}

应用变换法则: if

L {f (t)} = F (s)

则

L {e^{a t} f (t)} = F (s - a)

对于

e^{2 t} (t - 1)^{2} : f (t) = (t - 1)^{2}, a = 2

= L {(t - 1)^{2}} (s - 2)

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

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

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

化简

\frac{2}{( s - 2 ) ^{3}} - \frac{2}{( s - 2 ) ^{2}} + \frac{1}{s - 2} : \frac{s ^{2} - 6 s + 10}{( s - 2 ) ^{3}}

= \frac{s ^{2} - 6 s + 10}{( s - 2 ) ^{3}}

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