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I actually made a similar thread here: Solve this homogeneous type equation

and a user pointed out a mistake in my workings, however i could still not manage to get the solution. So I was wondering if someone could help with the last few parts

Question:

**2xyy' = y^2 - x^2**

y' = (y^2-x^2)/(2xy)

divide through by x^2 to get:

y' = ((y/x)^2-1)/(2(y/x)) = f(y/x)

let y/x = v, thus y = v + x'v

1/x dx = 1/(f(v)-v) dv

f(v)-v= (v^2-1)/2v - v

= (-2v^2+v^2-1)/2v

= -(v^2+1)/2v

integrating:

int(dx/x) = int(2v/-(v^2+1))

ln|x| + c = -ln|v^2+1|

e^(ln(x)+ C)= e^C e^(ln(x))= cx where c= e^C

and, finally, up to the point where I am stuck:

cx= -(v^2+ 1)

The solutions say

x^2+y^2=cx

I am unsure how to get to this, any help is appreciated.