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A Different Differential Equation

Age 16 to 18
Challenge Level

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?