Differential Equations/Formation of differential equations/Examples

Let us suppose we wanted to form a differential equation from

$y=a{x}^2+bx.$

We could take the first and second derivatives to get

$\frac{dy}{dx}=2ax+b$

and

$\frac{d^{2}y}{d{x}^2}=2a.$

Substituting for $2a$ in the first derivative we get

$\frac{dy}{dx}=x\frac{d^{2}y}{d{x}^2}+b$

and after gathering deriviates to left side we have

$\frac{dy}{dx}-x\frac{d^{2}y}{d{x}^2}=b.$

Substituting for $a$ and $b$ in the original equation and simplifying we see

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

This is a differential equation of the second order.