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I am trying to solve the equation below $$4xy'=4x^2y^2-4y-1$$

So far I am having a hard time in spotting a specific solution which I can use to find the general solution.

Any ideas?

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The general solution is $$ -\frac{\tanh \left(\frac{x}{2}-i c\right)}{2 x} $$ To get it change the variable $z(x)=x y(x)$ after that it is standard

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  • $\begingroup$ how did you come up with $z(x)=xy(x)$? $\endgroup$
    – Wanderer
    Sep 17, 2015 at 16:20
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    $\begingroup$ by looking on the equation it is clear that $x$ always comes together with $y$ $\endgroup$
    – Nikolay Gromov
    Sep 17, 2015 at 16:21
  • $\begingroup$ A specific solution would be $1\over 2x$ $\endgroup$
    – Wanderer
    Sep 17, 2015 at 16:41

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