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How to solve this differential equation:

$$x\frac{dy}{dx} = y + x\frac{e^x}{e^y}?$$

I tried to rearrange the equation to the form $f\left(\frac{y}{x}\right)$ but I couldn't thus I couldn't use $v = \frac{y}{x}$ to solve it.

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    This differential equation is not homogeneous (and so you can't rearrange it in the form $f(y/x)$.)2012-10-06
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    what is it then? how can you solve it?2012-10-06
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    If you assume $y>0$, something like $u=\log y$ may help. I haven't checked, so it may also be a waste of time.2012-10-06
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    The clue is that $\dfrac{e^x}{e^y}$ should rewrite to $e^{x-y}$ and thus we get the idea that let $u=x-y$ or let $u=y-x$ will convert the ODE whose the terms are not contain any composite functions.2012-10-07
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    The substitution $u = e^y$ leads to $$\frac{du}{dx} = \frac{dy}{dx}e^y = \frac yx e^y + e^x = \frac ux \log u + e^x.$$ May be this could lead somewhere.2012-10-07

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