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{\bf Question}

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Find the general solution of the equation $$ t \frac{dx}{dt} = x +
t e^{ \frac{x}{t}}$$

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{\bf Answer}

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$\displaystyle t \frac{dx}{dt} = x + t e^{ \frac{x}{t}} \ \ \
\Rightarrow \ \  \frac{dx}{dt} = \frac{x}{t} + e^{\frac{x}{t}}$

This is of the form $\frac{dx}{dt} = f\left( \frac{x}{t} \right)$
So let $y = \frac{x}{t} \Rightarrow \frac{dx}{dt} = t
\frac{dy}{dt} + y $

So we can rewrite as
\begin{eqnarray*}
t\frac{dy}{dt} + y & = & y + e^y \\ t\frac{dy}{dt}  & = & e^y
\\\int e^{-y} \, dy & = & \int \frac{dt}{t} \\ \Rightarrow -e^{-t}
& = & \ln |t| + \mathrm{constant} \\ t & = & Ae^{- (e^{-y})}
\end{eqnarray*}



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