laplacetransform e^{5(t-2)}(t^3+2t)^2

解

ラプラス変換

e^{5 (t - 2)} (t^{3} + 2 t)^{2}

\frac{720}{e ^{10} ( s - 5 ) ^{7}} + \frac{96}{e ^{10} ( s - 5 ) ^{5}} + \frac{8}{e ^{10} ( s - 5 ) ^{3}}

解答ステップ

L {e^{5 (t - 2)} (t^{3} + 2 t)^{2}}

拡張

e^{5 (t - 2)} (t^{3} + 2 t)^{2} : e^{5 t - 10} t^{6} + 4 e^{5 t - 10} t^{4} + 4 e^{5 t - 10} t^{2}

= L {e^{5 t - 10} t^{6} + 4 e^{5 t - 10} t^{4} + 4 e^{5 t - 10} t^{2}}

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 {e^{5 t - 10} t^{6}} + 4 L {e^{5 t - 10} t^{4}} + 4 L {e^{5 t - 10} t^{2}}

L {e^{5 t - 10} t^{6}} : \frac{720}{e ^{10} ( s - 5 ) ^{7}}

L {e^{5 t - 10} t^{4}} : \frac{24}{e ^{10} ( s - 5 ) ^{5}}

L {e^{5 t - 10} t^{2}} : \frac{2}{e ^{10} ( s - 5 ) ^{3}}

= \frac{720}{e ^{10} ( s - 5 ) ^{7}} + 4 \cdot \frac{24}{e ^{10} ( s - 5 ) ^{5}} + 4 \cdot \frac{2}{e ^{10} ( s - 5 ) ^{3}}

改良

\frac{720}{e ^{10} ( s - 5 ) ^{7}} + 4 \frac{24}{e ^{10} ( s - 5 ) ^{5}} + 4 \frac{2}{e ^{10} ( s - 5 ) ^{3}} : \frac{720}{e ^{10} ( s - 5 ) ^{7}} + \frac{96}{e ^{10} ( s - 5 ) ^{5}} + \frac{8}{e ^{10} ( s - 5 ) ^{3}}

= \frac{720}{e ^{10} ( s - 5 ) ^{7}} + \frac{96}{e ^{10} ( s - 5 ) ^{5}} + \frac{8}{e ^{10} ( s - 5 ) ^{3}}