Family of curves: parabola?

Question

Can you please help me understand the steps in finding the differential equation for parabolas whose vertex are on the y axis and with fixed focal distance a.

Someone said it is a parabolic PARTIAL DIFF EQ. But I dont know what that means

Answer 1

If the parabola is of the form y^2 = 4ax 

Then the differential is 

2ydy/dx = 4a 

dy/dx = 4a / 2y  = 2a / y