A difficulty in understanding a solution of a differential equation in Vinberg section 0.

Question

The differential equation is given in the following picture:

I believe that it is a separable differential equation, but because we differentiate with respect to you, I believe that the solution should be $$f(t) = C e^{ua}$$ instead of $$f(t) = C e^{ta},$$ could anyone clarify this for me please?

Answer 1

We have $ \frac{d}{du}f(t+u)=\frac{d}{du}f(t)f(u)$, hence $f'(t+u)=f(t)f'(u)$. With $u=0$ we get

$$f'(t)=f(t)f'(0).$$