The differential equation of the family of parabolas with focus at the origin and the x-axis as axis is
[EAMCET 2003]
−y(dydx)2+2xdydx=−y
Equation of family of parabolas with focus at
(0,0) and x - axis as axis is y2=4a(x+a)......(i)
Differentiating (i) with respect to x,
2yy1=4a;y2=2yy1(x+yy12)y=2xy1+yy21⇒y(dydx)2+2xdydx=y.