Question

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

Answer 1

The equation of the parabola having the vertex at origin and the axis along the positive y-axis is:

x2=4ay...(1)

Differentiating equation (1) with respect to x, we get:

2x=4ay'...(2)

Dividing equation (2) by equation (1) ,we get:

$\frac{2x}{x^2}=\frac{4ay'}{4ay}$

$\Rightarrow \frac{2}{x}=\frac{y'}{y}$

$\Rightarrow xy'=2y$

$\Rightarrow$ xy'-2y=0

This is the required differential equation.