Let M denote the family of parabolas whose vertex and focus both on the x-axis and let (a,0) be the focus of a member of the given family, where a is an arbitrary constant. Therefore, equation of family M is \(\displaystyle{y}^{{2}}={4}{a}{x}\)...(1)
Differentiating both sides of equation with respect to x, we get

\(\displaystyle{2}{y}{\left(\frac{{\left.{d}{y}\right.}}{{\left.{d}{x}\right.}}\right)}={4}{a}{x}\)

Substituting the value of 4a from equation in (1) \(\displaystyle{y}^{{2}}={\left({2}{y}{\left(\frac{{\left.{d}{y}\right.}}{{\left.{d}{x}\right.}}\right)}\right)}{x}\)

\(\displaystyle\to{y}^{{2}}={2}{x}{y}{\left(\frac{{\left.{d}{y}\right.}}{{\left.{d}{x}\right.}}\right)}\)

\(\displaystyle\to{y}^{{2}}-{2}{x}{y}{\left(\frac{{\left.{d}{y}\right.}}{{\left.{d}{x}\right.}}\right)}=-\)

which is the Differential equations of the family of parabolas.

\(\displaystyle{2}{y}{\left(\frac{{\left.{d}{y}\right.}}{{\left.{d}{x}\right.}}\right)}={4}{a}{x}\)

Substituting the value of 4a from equation in (1) \(\displaystyle{y}^{{2}}={\left({2}{y}{\left(\frac{{\left.{d}{y}\right.}}{{\left.{d}{x}\right.}}\right)}\right)}{x}\)

\(\displaystyle\to{y}^{{2}}={2}{x}{y}{\left(\frac{{\left.{d}{y}\right.}}{{\left.{d}{x}\right.}}\right)}\)

\(\displaystyle\to{y}^{{2}}-{2}{x}{y}{\left(\frac{{\left.{d}{y}\right.}}{{\left.{d}{x}\right.}}\right)}=-\)

which is the Differential equations of the family of parabolas.