laplacetransform x*sin(Ax)

解答

拉普拉斯变换

x \cdot sin (A x)

\frac{2 A s}{( s ^{2} + A ^{2} ) ^{2}}

求解步骤

L {x sin (A x)}

使用拉普拉斯变换表:

L {t^{k} f (t)} = (- 1)^{k} \frac{d ^{k}}{d s ^{k}} (L {f (t)})

对于

x sin (A x) : f (x) = sin (A x), k = 1

= (- 1)^{1} \frac{d}{d s} (L {sin (A x)})

L {sin (A x)} : \frac{A}{s ^{2} + A ^{2}}

\frac{d}{d s} (\frac{A}{s ^{2} + A ^{2}}) = - \frac{2 A s}{( s ^{2} + A ^{2} ) ^{2}}

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

= \frac{2 A s}{( s ^{2} + A ^{2} ) ^{2}}