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How do I solve the following ODE implicitly with the substitution $v=\frac xu$? $$\dfrac{dx}{du}=\frac{x+u}u$$

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Assume that $x=x(u)$ then substituting $$v=\frac{x}{u}$$ we get $$x' =v+v'u$$ so we obtain a equation with separates variables of the form $$v' u =1$$ hence $$v=\ln |u| +C$$ thus $$x(u) =u\ln |u| +Cu$$

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Here's how I would begin: $$\frac{x+u}{u}=\frac{x}{u}+\frac{u}{u}=\frac{x}{u}+1=v+1$$

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