I was wondering on the following and I probably know the answer already: NO.
Is there another number with similar properties as e? So that the derivative of exp(x) is the same as the function itself.
I can guess that it’s probably not, because otherwise e wouldn’t be that special, but is there any proof of it?
Of course Cex has the same property for any C (including C=0). But these are the only ones.
Proposition: Let f:R→R be a differentiable function such that f(0)=1 and f′(x)=f(x). Then it must be the case that f=ex.
Proof. Let g(x)=f(x)e−x. Then
by assumption, so g is constant. But g(0)=1, so g(x)=1 identically.
N.B. Note that it is also true that ex+c has the same property for any c. Thus there exists a function g(c) such that ex+c=g(c)ex=ecg(x), and setting c=0, then x=0, we conclude that g(c)=ec, hence ex+c=exec.
This observation generalizes to any differential equation with translation symmetry. Apply it to the differential equation f″ and you get the angle addition formulas for sine and cosine.