What is laplacetransform t*cos(pi*t) ?

The answer to laplacetransform t*cos(pi*t) is -(-s^2+pi^2)/((s^2+pi^2)^2)

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laplacetransform tcos(pit)

laplace transform

t \cdot cos (π \cdot t)

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

Solution steps

L {t cos (π t)}

Use Laplace Transform table:

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

For

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}}

y^{^{'}} + 2 y = 3 sin (e^{2 x})

n = 1 \sum \infty \frac{( 2 ^{2 n} n ! )}{2 n + 1}

e^{- 3 t} \cdot sin (t)

x \to 3 + lim (\frac{2}{x + 3})

F (x) = x^{4} - 8 x^{2} + 2

What is laplacetransform t*cos(pi*t) ?
The answer to laplacetransform t*cos(pi*t) is -(-s^2+pi^2)/((s^2+pi^2)^2)