Find the general integral of the homogeneous equation…
Find the general integral of the homogeneous equation $x^2+y^2-2xyy'=0$.
Explanation of Method
Solve for $y'$ to see it depends only on $y/x$, substitute $y=vx$, integrate the resulting separable equation, and simplify the implicit result to a family of rectangular hyperbolas.
Step 1: Substitute $y=vx$
From $y'=\dfrac{x^2+y^2}{2xy}=\dfrac{1+v^2}{2v}$ with $y=vx$, $y'=v+xv'$, so
\[x\frac{dv}{dx}=\frac{1+v^2}{2v}-v=\frac{1-v^2}{2v}.\]
Step 2: Separate and integrate
\[\frac{2v\,dv}{1-v^2}=\frac{dx}{x} \quad\Longrightarrow\quad -\ln|1-v^2|=\ln|x|+C \quad\Longrightarrow\quad 1-v^2=\frac{K}{x}.\]
Step 3: Back-substitute $v=y/x$
\[1-\frac{y^2}{x^2}=\frac{K}{x} \quad\Longrightarrow\quad \boxed{\,x^2-y^2=Cx.}\]
Result
Final answer:
\[\boxed{\,x^2-y^2=Cx\ \ \Longleftrightarrow\ \ \left(x-\dfrac{C}{2}\right)^2-y^2=\dfrac{C^2}{4}}\]
— a family of rectangular hyperbolas.