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Differentiate w.r.t $x$:

$$x+y\frac{dx}{dy}=4c^2$$

amWhy
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  • Your differentiation is wrong. The derivative of a constant is 0 and you want $\frac{dy}{dx}$, not $\frac{dx}{dy}$. You should have $x+ y\frac{dy}{dx}= 0$. From that $\frac{dy}{dx}= -\frac{x}{y}$ and the orthogonal complement will satisfy $\frac{dy}{dx}= \frac{y}{x}$. – user247327 Dec 06 '18 at 12:33
  • I would argue the derivative w.r.t $x$ equals $2x + y \frac{\delta y}{\delta x} = 0$, thus $\frac{\delta y}{\delta x}=\frac{-2x}{y}$. Hope this helps. – Mathbeginner Dec 06 '18 at 12:44

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$$x+yy'=0$$ turns to

$$x-\frac y{y'}=0$$ which integrates as $$y=cx.$$

This was expected as the given family is made of circles centered at the origin.