laplacetransform 2e^{2t}+8te^{2t}+4t^2e^{2t}

解答

拉普拉斯变换

2 e^{2 t} + 8 t e^{2 t} + 4 t^{2} e^{2 t}

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

求解步骤

L {2 e^{2 t} + 8 e^{2 t} t + 4 e^{2 t} t^{2}}

利用拉普拉斯变换的线性特性:

对于函数

f (t), g (t)

和常数

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

= 2 L {e^{2 t}} + 8 L {t e^{2 t}} + 4 L {t^{2} e^{2 t}}

L {e^{2 t}} : \frac{1}{s - 2}

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

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

= 2 \cdot \frac{1}{s - 2} + 8 \cdot \frac{1}{( s - 2 ) ^{2}} + 4 \cdot \frac{2}{( s - 2 ) ^{3}}

化简

2 \frac{1}{s - 2} + 8 \frac{1}{( s - 2 ) ^{2}} + 4 \frac{2}{( s - 2 ) ^{3}} : \frac{2 s ^{2}}{( s - 2 ) ^{3}}

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

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