<s>[INST] Solving $y'=\frac{1 + y^2 - x^2}{2 x y}$ I need help to solve the equation
$$y'=\frac{1 + y^2 - x^2}{2 x y}$$
Can I transform it into a homogeneous equation?
Actually this problem came from Apostol's Calculus I, 8.28.3

Find the orthogonal trajectories of $x^2 + y^2 + 2 C y = 1$.

So it is better not to use anything beyond Chapter 8.
 [/INST] $$\begin{align}
\frac{dy}{dx} &= \frac{1+y^2-x^2}{2xy}
\\2xydy &= dx+ y^2dx-x^2dx
\\x^2dx &=y^2dy-2xydy+dx
\\\frac{y^2dx-2xydy}{x^2}&=dx-\frac{dx}{x^2}
\\\frac{y^2dx-xd(y^2)}{x^2}&=dx-\frac{dx}{x^2}
\\-d\left(\frac{y^2}{x}\right)&=dx-\frac{dx}{x^2}
\end{align}$$
Now integrating,
$$-\frac{y^2}{x} = x+\frac{1}{x}+c
\\y^2+x^2+cx=-1
$$
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