(adsbygoogle = window.adsbygoogle || []).push({}); First Order Linear Non-Homogeneous Equation

1. The problem statement, all variables and given/known data

I need to solve for e(t)

2. Relevant equations

Do I use Laplace Transform for the last integral?

3. The attempt at a solution

[tex]

\begin{subequations}

\begin{eqnarray}

\nonumber

\dot{\hat{{\cal E}}}(t) &=& -\kappa \hat{{\cal E}}(t) + \sqrt{2\kappa}\, \hat{{\cal E}}_{in}(t), \\

\end{eqnarray}

\end{subequations}

\begin{subequations}

\begin{eqnarray}

\dot{\hat{{\cal E}}} &=& -\kappa \hat{{\cal E}} + \sqrt{2\kappa}\, \hat{{\cal E}}_{in}, \\

\nonumber

I &=& e^{\int \kappa dt} =ce^{\kappa t}\\

\nonumber

ce^{\kappa t}\dot{\hat{{\cal E}}} + ce^{\kappa t}\kappa \hat{{\cal E}} &=& ce^{\kappa t} \sqrt{2\kappa}\, \hat{{\cal E}}_{in}, \\

\nonumber

\frac{d}{dt}(\hat{{\cal E}}e^{ \kappa t}) &=& ce^{\kappa t} \sqrt{2\kappa}\, \hat{{\cal E}}_{in}\\

\nonumber

\hat{{\cal E}}e^{ \kappa t} &=& \int^{\infty}_{0}ce^{\kappa t} \sqrt{2\kappa}\, \hat{{\cal E}}_{in}(t)dt\\

\nonumber

\nonumber

\end{eqnarray}

\end{subequations}

[/tex]

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# Homework Help: First Order Linear Homogeneous Equation

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