transform of derivative

Problem 04
Find the Laplace transform of   $f(t) = t \, \sin t$   using the transform of derivatives.
 

Solution 04
$f(t) = t \, \sin t$       ..........       $f(0) = 0$

$f'(t) = t \, \cos t + \sin t$       ..........       $f'(0) = 0$

$f''(t) = (-t \, \sin t + \cos t) + \cos t = -t \, \sin t + 2\cos t$
 

$\mathcal{L} \left\{ f''(t) \right\} = s^2 \mathcal{L} \left\{ f(t) \right\} - s \, f(0) - f'(0)$

Problem 03
Find the Laplace transform of   $f(t) = e^{5t}$   using the transform of derivatives.
 

Solution 03
$f(t) = e^{5t}$       ..........       $f(0) = 1$

$f'(t) = 5e^{5t}$
 

$\mathcal{L} \left\{ f'(t) \right\} = s \, \mathcal{L} \left\{ f(t) \right\} - f(0)$

$\mathcal{L} (5e^{5t}) = s \, \mathcal{L} (e^{5t}) - 1$

$1 = s \, \mathcal{L} (e^{5t}) - \mathcal{L} (5e^{5t})$

$s \, \mathcal{L} (e^{5t}) - 5 \, \mathcal{L} (e^{5t}) = 1$

Problem 02
Find the Laplace transform of   $f(t) = \sin^2 t$   using the transform of derivatives.
 

Solution 02
$f(t) = \sin^2 t$       ..........       $f(0) = 0$

$f'(t) = 2\sin t \, \cos t = \sin 2t$
 

$\mathcal{L} \left\{ f'(t) \right\} = s \, \mathcal{L} \left\{ f(t) \right\} - f(0)$

$\mathcal{L} (\sin 2t) = s \, \mathcal{L} (\sin^2 t) - 0$

$\dfrac{2}{s^2 + 2^2} = s \, \mathcal{L} (\sin^2 t)$

$s \, \mathcal{L} (\sin^2 t) = \dfrac{2}{s^2 + 4}$

Problem 01
Find the Laplace transform of   $f(t) = t^3$   using the transform of derivatives.
 

Solution 01
$f(t) = t^3$       ..........       $f(0) = 0$

$f'(t) = 3t^2$       ..........       $f'(0) = 0$

$f''(t) = 6t$       ..........       $f''(0) = 0$

$f'''(t) = 6$
 

$\mathcal{L} \left\{ f'''(t) \right\} = s^3 \mathcal{L} \left\{ f(t) \right\} - s^2 f(0) - s \, f'(0) - f''(0)$