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

解

ラプラス変換

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

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

解答ステップ

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

変換ルールを適用:

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

ならば

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

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

向け

: 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}

簡素化

\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}}