Prove: If m is any root of f(m) = 0 , then

            f(D)e^mx = 0 

Proof. Let f(D) be a polynomial in D

            f(D) = a₀Dⁿ + a₁D^n-1 + ....... + a_n-1D + a_n

Then, because D^ke^mx = m^ke^mx, we have

            f(D)e^mx = a₀mⁿe^mx + a₁m^{n -1}e^mx + ........ + a_{n -1} me^mx + a₀e^mx

or

1)        f(D)e^mx = e^mxf(m)

From 1) we see that if m is a root of the equation f(m) = 0, then f(D)e^mx = 0.