Differential Equations/Formation of differential equations/Examples

Current revision (unreviewed)

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

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

Substituting for $2a\,\!$ in the first derivative we get

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

and after gathering deriviates to left side we have

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

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

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

This is a differential equation of the second order.