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

解答

拉普拉斯变换

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

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

求解步骤

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

展开

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

= L {2 e^{2 t} t - 2 e^{t} t + t^{2} + e^{4 t} - 2 e^{3 t} + e^{2 t}}

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} t} - 2 L {e^{t} t} + L {t^{2}} + L {e^{4 t}} - 2 L {e^{3 t}} + L {e^{2 t}}

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

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

L {t^{2}} : \frac{2}{s ^{3}}

L {e^{4 t}} : \frac{1}{s - 4}

L {e^{3 t}} : \frac{1}{s - 3}

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

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

整理

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

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

d er i v a t i v e 2 x^{2} - 7 x - 195

\int - cos (\frac{π t}{4}) d t

x \to - 2 lim (- x^{2} + 6 x - 8)

\int a arctan (b x) d x

x \to 1 lim (x^{2} + 4)