Functions that are always less than their derivatives

Solution 1:

If $y'(x)>y(x)\quad\forall x\in\mathbb{R}$, we can define $f(x)=y'(x)-y(x)$ which is positive forall $x$. Suppose that $y'(x)$ is continuous function so that $f(x)$ is continuous too. Now with this element we can build the differential equation $$y'(x)=y(x)+f(x)$$ and its solutions are given by: $$y(x)=e^{x}\left(c+\int_{x_0}^{x}e^{-s}f(s)ds\right)$$

Again are there any applications/physical manifestations of these functions? [for example an object with a velocity that is always greater than its position/acceleration is always greater than its velocity]

I don't know if there's application of this interesting property, but I'm sure that you can't compare velocity with the position because they are not homogeneous quantities.

Solution 2:

Assuming $f(x)>0$, $f:\mathbb{R}\mapsto\mathbb{R}$

$f'(x) > f(x) \iff \frac{d}{dx}\ln(f(x))>1$

So you can turn any function $g$ where $g'(x)>1$ into this type of function by taking the exponential of it:

$\frac{d}{dx}g(x)>1 \implies \frac{d}{dx}\ln(e^{g(x)})>1 \implies \frac{d}{dx} e^{g(x)}>e^{g(x)}$

Solution 3:

A simple example is $f(x)=-x^2-3$