I can't seem to solve this question, please tell me what I am doing wrong:

q1.

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 + xv'

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|

x + c = -(v^2-1)

x + v^2 -1 = c

v = +-(c-x+1)

y = xv

y = +-x(c-x+1)

..

fairly sure i am correct up until i take the logs out.

The answer book says the answer is

x^2 + y^2 = cx

I am unsure how to get this.