I How to solve an ODE to find its solution

Salutations, I have a problem when I approach this ODE:

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

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.
 

Delta2

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Well this seems like a hard non linear ODE. I tried it at wolfram and the solution seems quite complex and it is given in implicit (perplexed form) ##G(y,x)=0##

I can comment only on one of your result, the ##y=kx## simply does not verify the ODE, cause in the left hand side we get ##x^2(k^2+1)## while in the right hand side we get 0.

You can check the solution of wolfram here https://www.wolframalpha.com/input/?i=solve+ODE+(y/y')^2+y^2=b^2(x-(y/y'))^2 but you cant see the step by step solution unless you are a subscribed user at wolfram.
 
Thanks for your commentary @Delta2 about wolfram it's true, I'm not premium user, but thanks for it, indeed, y=kx is not solution, thanks for the link again.
 

phyzguy

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y=kx is a solution if k=+/-i. In other words, y = i*x or y = -i*x are solutions.
 

Delta2

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y=kx is a solution if k=+/-i. In other words, y = i*x or y = -i*x are solutions.
That is true, however in his/her workings he writes ##y=kx, k\in\mathbb{R}##, and i also thought we are interested for real valued functions y.
 

phyzguy

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That is true, however in his/her workings he writes ##y=kx, k\in\mathbb{R}##, and i also thought we are interested for real valued functions y.
Sorry, I missed that.
 
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Hi, your ode splits into two odes:
##y' = \frac{(y*\sqrt((b^2-1)*y^2+b^2*x^2)-b^2*x*y)}{(y^2-b^2*x^2)}##
##y' = -\frac{(y*\sqrt((b^2-1)*y^2+b^2*x^2)+b^2*x*y)}{(y^2-b^2*x^2)}##
an integrating factor for the first one is
##\frac{1}{(b^2*(b^2-2)*y*(x*\sqrt((b^2-1)*y^2+b^2*x^2)-y^2))}##
and the second one is
##\frac{1}{(b^4*y*(x*\sqrt((b^2-1)*y^2+b^2*x^2)+y^2))}##

When trying to find the solution from this integrating factor, I couldn't get it to simplify to something readable...
I only managed to find the integrating factor by assuming that the ODE has a generalized rational point symmetry using Lie's symmetry method.
 

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