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I have an equation of the form:

$\frac {(y')^2}{2} +xy' = y$

I realize it's a Clairaut and now I derive each term:

$y'=y'+xy''+ \frac {d\frac {(y')^2}{2}}{dx}$

I dont feel quite confident of my following differentiation. I know the result is correct, but is there any algebraic error in the following steps?

$u=f'(x)$

$du=f''(x)dx$

$\frac {du}{f''(x)}=dx$

$\frac {d\frac {(y')^2}{2}}{dx}=y''(x)\frac{d}{du}\frac{u^2}{2}=y''(x)y'(x)$

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    Wait,...what's $u$ doing in there? And what does the "du" sitting alone in the last line mean? This looks pretty muddled, and it's just a simple chain-rule application. – 2017-02-15
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    Damn, didnt check twice when writing, edited last row. – 2017-02-15
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    you are correct there. – 2017-02-15

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