ODE6.2

Find the Laplace transform of $e^{at}\,$

$L\{e^{at}; p\} = \int_0^\infty e^{-pt}e^{at}\,dt\,$

$= \int_0^\infty e^{-(p-a)t}\,dt\,$

$= \lim_{b\rightarrow\infty} \frac{e^{-(p-a)t}}{-(p-a)}\Bigg|_{t=0}^b = \frac{1}{p-a}\,$

This integral diverges for $p<a\,$ .

Let $a\rightarrow 0^+\,$ to get $L\{1; p\} = \frac{1}{p}\,$