laplacetransform cos^2(21t)

解答

拉普拉斯变换

cos^{2} (21 t)

\frac{1}{2 s} + \frac{s}{2 ( s ^{2} + 1764 )}

求解步骤

L {cos^{2} (21 t)}

使用三角恒等式改写

L {\frac{1}{2} + \frac{1}{2} cos (42 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)}

= L {\frac{1}{2}} + \frac{1}{2} L {cos (42 t)}

L {\frac{1}{2}} : \frac{1}{2 s}

L {cos (42 t)} : \frac{s}{s ^{2} + 1764}

= \frac{1}{2 s} + \frac{1}{2} \cdot \frac{s}{s ^{2} + 1764}

整理

\frac{1}{2 s} + \frac{1}{2} \frac{s}{s ^{2} + 1764} : \frac{1}{2 s} + \frac{s}{2 ( s ^{2} + 1764 )}

= \frac{1}{2 s} + \frac{s}{2 ( s ^{2} + 1764 )}

a re a 4 x^{2} - 4, x^{4} - 1

x \to \infty lim (- \frac{9}{x})

\int (t e^{- t}) d t

\frac{d y}{d x} = sin (2 x)

a re a y^{2} = x + 1, y + x = 1