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I have a question concerning the method for solving the following ODE: Let $y = f(x)$ satisfying $$\left( \frac{dy}{dx} \right)^2 + x \frac{dy}{dx} \frac{d^2 y}{dx^2} = 1.$$

I cannot seem to find a general method for solving such equations, at least in the standard references I have. I would be happy with a reference for this question, but an answer is also welcome. I apologise in advance if a question of this form has already been made available here.

Side Remark: The above equation arises in the study of collapsing Ricci-flat metrics.

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    $\begingroup$ Try defining a new variable $u = \frac{dy}{dx}$. At least you reduce the order. $\endgroup$ – Gonzalo Benavides Jun 12 '19 at 1:02
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Let $z=(y')^2$ then the differential equation is $$xz'+2z=2$$ which is a first order DE.

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  • $\begingroup$ Thankyou very much. Very easy! I'll accept your answer as soon as I can. $\endgroup$ – AmorFati Jun 12 '19 at 1:04
  • $\begingroup$ You are welcome :) $\endgroup$ – Nosrati Jun 12 '19 at 1:07

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