Prove that there is no p P3 such that px px for all x and p

Prove that there is no p P_3 such that p\'(x) = p(x) for all x and p(0) = 1. Philosophical note: the exponential function is not a polynomial, and one could argue that this question shows that the exponential function is a rather interesting object.

Solution

Normally if p\'(x) = p(x) then general solution is

p(x) = e^x with p(0) = 1

But since here it is given a polynomial funciton

Let p(x) = a0+a1x+a2x²+...+anxⁿ+...

and if possible let p(x) = p\'(x)

Then p\'(x) = a1+2a2x+...+nanx^n-1+... = a0+a1x+a2x²+...+anxⁿ+...

This implies na_n = (n+1)a_n+1

p(0) = 1 gives a0 = 1

Hence we get a1 = 1 = a2=a3=a4=...

But P(x) is only of degree 3 where as we got solution as infinite degree polynomial.

Contradiction.

Hence there cannot be any polynomial of degree 3, such that p(x) = p\'(x) and p(0) =1