Question

Form the differential equation of all parabolas whose axis is $y$-axis.

Answer 1

Equation of parabola with y-axis as vertex : $y=a{x}^2+b$

$\frac{dy}{dx}=2ax$ ___ (1), $\frac{d^{2}y}{d{x}^2}=2a$ ___ (2)

substituting (2) in (1), $\frac{dy}{dx}=x\frac{d^{2}y}{d{x}^2}$

$x\frac{d^{2}y}{d{x}^2}-\frac{dy}{dx}=0$ is the required differential equation.