This is an investigation concerning differential equations -- it is both difficult and open-ended. You might wish to discuss the problem in groups and make good use of computers or other tools to get started before closing the problem down to explore a particular route.
 
 
Can you find any solutions to a differential equation of the form
 
$$
f'(x)= cf(x^2)\quad\mbox{ or } \quad f'(x^2)=cf(x)\,,
$$
where $c$ is a constant?