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$$\left(\frac{y}{y'}\right)^2+y^2=b^2\left(x-\frac{y}{y'}\right)^2$$

I have done a series of steps as I show in this link:

https://drive.google.com/file/d/1Ht4xxUlm7vXqg4S5-wirKwm7vTESU3mU/view?usp=sharing

But I'm not convinced that those were the correct steps neither solutions were adequated, and my question is:

How would be the mathematical steps to apply to find the correct solution of the ODE?

So, I would like any guidance or starting steps or explanations to find the solution of this interesting problem.

Thanks for your attention.