# Question # Obtain the Differential equations: parabolas with vertex and focus on the x-axis.

Differential equations
ANSWERED Obtain the Differential equations: parabolas with vertex and focus on the x-axis. 2020-11-11
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.