laplacetransform (sin(6t)-7)^2

解答

拉普拉斯变换

(sin (6 t) - 7)^{2}

\frac{1}{2 s} - \frac{s}{2 ( s ^{2} + 144 )} - \frac{84}{s ^{2} + 36} + \frac{49}{s}

求解步骤

L {(sin (6 t) - 7)^{2}}

展开

(sin (6 t) - 7)^{2} : sin^{2} (6 t) - 14 sin (6 t) + 49

= L {sin^{2} (6 t) - 14 sin (6 t) + 49}

使用三角恒等式改写

L {\frac{1}{2} - \frac{1}{2} cos (12 t) - 14 sin (6 t) + 49}

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 (12 t)} - 14 L {sin (6 t)} + L {49}

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

L {cos (12 t)} : \frac{s}{s ^{2} + 144}

L {sin (6 t)} : \frac{6}{s ^{2} + 36}

L {49} : \frac{49}{s}

= \frac{1}{2 s} - \frac{1}{2} \cdot \frac{s}{s ^{2} + 144} - 14 \cdot \frac{6}{s ^{2} + 36} + \frac{49}{s}

整理

\frac{1}{2 s} - \frac{1}{2} \frac{s}{s ^{2} + 144} - 14 \frac{6}{s ^{2} + 36} + \frac{49}{s} : \frac{1}{2 s} - \frac{s}{2 ( s ^{2} + 144 )} - \frac{84}{s ^{2} + 36} + \frac{49}{s}

= \frac{1}{2 s} - \frac{s}{2 ( s ^{2} + 144 )} - \frac{84}{s ^{2} + 36} + \frac{49}{s}

\int \frac{cos ^{2} ( θ )}{sin ( θ )} d θ

d er i v a t i v e f (x) = 2^{90}

x \to 5 + lim (x^{3} - c x)

2 y^{^{''}} - 4 y^{^{'}} + 3 y = 9 t^{2} + 10

\frac{d}{d x} (3^{3 x + 1})