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__EDIT__: my problem is solved, thank you to those who helped

## Homework Statement

Solve:

[itex]x y^{\prime \prime} = y^{\prime} \log (\frac{y^{\prime}}{x})[/itex]

__Note__: This is the first part of an undergraduate applications course in differential equations. We were taught to solve second order non-linear equations by doing tricks such as substituting [itex]u = y^{\prime}[/itex], factoring, and then dealing with the two cases that result.

## Homework Equations

None.

## The Attempt at a Solution

I tried [itex]u := y^{\prime}[/itex], so [itex]u^{\prime} = y^{\prime \prime}[/itex], and so substitution gives

[itex]x u^{\prime} = u \log(\frac{u}{x})[/itex]

but I can't seem to be able to do anything with that.

I also I noticed that

[itex] y^{\prime \prime} = y^{\prime} x^{-1} \log(y^{\prime} x^{-1}) [/itex]

which I thought was interesting. But I couldn't really do anything with it.

Thank you.

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