One function satisfies another, a bit confused

Hi there,
I'm given the following question:
We know that y(x) = e^x satisfies y'(x) = y(x). Given any real number 'a' find a function y(x) that satisfies y'(x) = ay(x).

I'm not sure how to go about it. I know that the derivative of e^x is e^x itself, but how could I then find ay(x)? Any guidance would be greatly appreciated. Thank you!

You have to remember the chain rule in dealing with e^f(x)

d/dx (e^f(x)) = e^f(x) d/dx f(x)

so you are looking for a = d/d/x f(x) take the antiderivative and ax = f(x) for all values of a

Any value for a will work as long as the function is e^ax
use a = 3

f(x) = e^3x
f'(x) = e^3x d/dx (3x) = e^3x (3) = 3e^3x

You could set a = 1/2 and do:

y(x) = 1/2*e^2x which would make

y'(x) = e^2x

y = 0 meets the condition.