The differential equation of the family of parabola with focus at the…

The equation of the family of parabolas with focus at the origin and the x-axis as axis is

$y^2=4 a(x-a)$, where $a$ is a parameter.         ......(i)

Differentiating with respect to x, we get

$2 y \frac{d y}{d x}=4 a \Rightarrow a=\frac{y}{2} \frac{d y}{d x}$

Substituting the value of $a$ in (i), we get

$y^2 =2 y \frac{d y}{d x}\left(x-\frac{y}{2} \frac{d y}{d x}\right)$

$\Rightarrow y^2 =y \frac{d y}{d x}\left(2 x-y \frac{d y}{d x}\right) \Rightarrow y\left(\frac{d y}{d x}\right)^2-2 x \frac{d y}{d x}+y=0$

This is the required differential equation.