Question

Form the differential equations for all parabolas whose axis is Y-axis.

Answer 1

Equation of parabola whose axis is along $y-$axis is

$x^2=4ay$ ....(i)

$\Rightarrow \frac{d}{dx}\left(x^2\right)=4a\frac{dy}{dx}$

$\Rightarrow 2x=\frac{x^2}{y}.\frac{dy}{dx}$

$\Rightarrow \frac{dy}{dx}=\frac{2xy}{x^2}$

$\therefore \frac{dy}{dx}=\frac{2y}{x}$

Which is Differential equation of parabola whose axis is along $y-$axis.