ODE6.2

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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}}\,

Ordinary Differential Equations

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