ENGR201 Laplace deck examples, transforms from the definition (ENGR5001) - Revisica

ENGR201 Laplace deck examples, transforms from the definitionTutorialOld spec ENGR2012:212 min

ENGR201 (old spec) Laplace transforms deck, "some examples" (pages 6-9), cosh example and "try L[sinh at]" (page 9)

Using the definition of the Laplace transform (and its linearity), find:

Formulas you may need

Definition $F(s) = \mathcal{L}[f(t)] = \displaystyle\int_0^\infty e^{-st}f(t)\,dt$ (learn this)
$\displaystyle\int e^{\beta t}dt = \dfrac{1}{\beta}e^{\beta t}$ ; $\cosh at = \dfrac{e^{at} + e^{-at}}{2}$ , $\sinh at = \dfrac{e^{at} - e^{-at}}{2}$ (learn this)
Linearity $\mathcal{L}[\alpha f + \beta g] = \alpha\mathcal{L}[f] + \beta\mathcal{L}[g]$ (learn this)

(a)
$\mathcal{L}[1]$ (for $t > 0$ )
(b)
$\mathcal{L}[e^{at}]$ , $a$ a real constant, stating the condition on $s$ ; hence $\mathcal{L}[e^{-at}]$
(c)
$\mathcal{L}[u(t - t_0)]$ , where $u(t - t_0)$ is the unit step starting at $t = t_0 > 0$
(d)
$\mathcal{L}[\cosh at]$
(e)
$\mathcal{L}[\sinh at]$ (the "try this" homework)