laplacetransform tcos(pit)

解答

拉普拉斯变换

t \cdot cos (π \cdot t)

- \frac{- s ^{2} + π ^{2}}{( s ^{2} + π ^{2} ) ^{2}}

求解步骤

L {t cos (π t)}

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

t cos (π t) : f (t) = cos (π t), k = 1

= (- 1)^{1} \frac{d}{d s} (L {cos (π t)})

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

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

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

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

l a pl a ce t r an s f or m e^{- 3 t} \cdot sin (t)

\frac{d}{d x} (sin (5) x^{5})

\frac{d}{d x} (x^{4 s i n (x)})

y d x + x (ln (x) - ln (y) - 1) d y = 0, y (1) = e

d er i v a t i v e 4 t^{2} - t