Question

dx? 1. f (x) = x2 and thus, as we will learn, f ′(x) = 2x. Find a differential equation for f . For this example, it should be of the form f ′(x) = f (x) ∗ g(x) where your task is to find g(x). g will not be defined at x = 0, but don’t worry about that right now.

Answer 1

This boils down to simple algebraic manipulation.

$f'(x)=2x\implies\dfrac{f'(x)}2=x\implies\dfrac{xf'(x)}2=x^2=f(x)$

So one possible ODE (the simplest, in fact) is

$f'(x)=\dfrac2x f(x)$