laplacetransform (2+t)^4

解

ラプラス変換

(2 + t)^{4}

\frac{16}{s} + \frac{32}{s ^{2}} + \frac{48}{s ^{3}} + \frac{48}{s ^{4}} + \frac{24}{s ^{5}}

解答ステップ

L {(2 + t)^{4}}

拡張

(2 + t)^{4} : 16 + 32 t + 24 t^{2} + 8 t^{3} + t^{4}

= L {16 + 32 t + 24 t^{2} + 8 t^{3} + t^{4}}

ラプラス変形の線形性を使用する:

関数

f (t), g (t)

と定数

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

= L {16} + 32 L {t} + 24 L {t^{2}} + 8 L {t^{3}} + L {t^{4}}

L {16} : \frac{16}{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}}

= \frac{16}{s} + 32 \cdot \frac{1}{s ^{2}} + 24 \cdot \frac{2}{s ^{3}} + 8 \cdot \frac{6}{s ^{4}} + \frac{24}{s ^{5}}

改良

\frac{16}{s} + 32 \frac{1}{s ^{2}} + 24 \frac{2}{s ^{3}} + 8 \frac{6}{s ^{4}} + \frac{24}{s ^{5}} : \frac{16}{s} + \frac{32}{s ^{2}} + \frac{48}{s ^{3}} + \frac{48}{s ^{4}} + \frac{24}{s ^{5}}

= \frac{16}{s} + \frac{32}{s ^{2}} + \frac{48}{s ^{3}} + \frac{48}{s ^{4}} + \frac{24}{s ^{5}}