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Math Help - differential equations

  1. #1
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    differential equations

    Laplace transform of (1-e^-t)/(t)
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  2. #2
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    Re: differential equations

    \displaystyle \begin{align*} \mathcal{L}\left\{ \frac{1 - e^{-t}}{t} \right\} &= \mathcal{L}\left\{ \frac{1}{t}\left( 1 - e^{-t}\right) \right\} \\ &= \int_s^{\infty}{\frac{1}{u} - \frac{1}{u + 1}\,du} \\ &= \lim_{\epsilon \to \infty}\int_s^{\epsilon}{\frac{1}{u} - \frac{1}{u + 1}\,du} \\ &= \lim_{\epsilon \to \infty}\left[\ln{\left|u\right|} - \ln{\left|u + 1\right|} \right]_s^{\epsilon} \\ &= \lim_{\epsilon \to \infty}\left[ \ln{\left| \frac{u}{u + 1} \right|} \right]_s^{\epsilon} \\ &=\lim_{\epsilon \to \infty} \left[\ln{\left| 1 - \frac{1}{u + 1} \right|}\right]_s^{\epsilon} \\ &= \lim_{\epsilon \to \infty}\ln{\left|1 -  \frac{1}{\epsilon + 1}  \right|} - \ln{\left|1 - \frac{1}{s + 1}\right|} \\ &= \ln{|1|} - \ln{\left|1 - \frac{1}{s + 1}\right|} \\ &= -\ln{\left| 1 + \frac{1}{s + 1} \right|} \end{align*}

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