Question

Form the differential equation of the family of parabolas having vertex at origin and axis along positive $y$-axis.

Answer 1

Equation of such parabola is given by $x^2=4ay$

In order to make a differential equation, we just have to remove $a$

Differentiating both the sides we get

$2x=4a{y}^{{}^{\prime }}$

$a=\frac{x}{2y^{{}^{\prime }}}$

Putting this in original equation of parabola, we get

$x{y}^{{}^{\prime }}-2y=0$