Finding an integral curve (applications of differential equations) Find a curve whose distance of every tangent from the origin $ON$ is equal to the $x$ axis coordinate of the point of intersection between the curve and that tangent $OU$.
How to set up the graph for this kind of problems in general?
How to form an ODE?
How to choose constant of integration and the resulting graph?
EDIT:
Here is the sketch:

Line $OB$ is orthogonal to tangent. The condition is that $OA=OB$

The equation of the tangent at abscissa $x$ is
$$Y-y(x)-y'(x)(X-x)=0$$ so that the distance to the origin is
$$\frac{|0-y-y'\cdot(0-x)|}{\sqrt{1+y'^2}}$$ and is known to equal $x$.
Squaring and rearranging,
$$y^2-2xyy'+x^2y'^2=x^2+x^2y'^2,$$
$$y^2-2xyy'=x^2,$$
$$\left(\frac{y^2}x\right)'=-1.$$
Then after integration
$$y^2=x(C-x).$$

Discussion:
As the distance is an absolute value, both signs in $y=\pm\sqrt{x(C-x)}$ are valid and we can leave the equation in its quadratic form.
