A different differential equation

Explore the properties of this different sort of differential equation.
Age
16 to 18
Challenge level
filled star filled star filled star
Exploring and noticing Working systematically Conjecturing and generalising Visualising and representing Reasoning, convincing and proving
Being curious Being resourceful Being resilient Being collaborative

Problem



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?