Problem 04
Find the Laplace transform of $f(t) = t \, \sin t$ using the transform of derivatives.
Solution 04
$f(t) = t \, \sin t$ .......... $f(0) = 0$
$f'(t) = t \, \cos t + \sin t$ .......... $f'(0) = 0$
$f''(t) = (-t \, \sin t + \cos t) + \cos t = -t \, \sin t + 2\cos t$
$\mathcal{L} \left\{ f''(t) \right\} = s^2 \mathcal{L} \left\{ f(t) \right\} - s \, f(0) - f'(0)$