Everything starts from the equation of the parabola. Parabolas with vertices and foci on the abscissa all have the following equation: where is the distance between the focus and the origin of the coordinate system (in this case its the distance along the abscissa). Applying derivative with respect to and you get the following differential equation: where you can substitute with general constant term providing the constraint since it carries the information about the distance which cannot be negative.

Hope this helps.