To prove the given identity, first we need to determine the terms in identity, namely f"(x) and f'(x).
We'll begin with f'(x):
f'(x) = (2e^2x-3e^-x)'
f'(x) = 4e^2x + 3e^-x
f"(x) = [f'(x)]'
f"(x) = (4e^2x + 3e^-x)
f"(x) = 8e^2x - 3e^-x
Now, we'll substitute the expressions of f"(x) andf'(x) into the identity that has to be verified:
8e^2x - 3e^-x - 4e^2x - 3e^-x - 2(2e^2x-3e^-x)
We'll remove the brackets and we'll combine like terms:
4e^2x - 6e^-x - 4e^2x + 6e^-x
We'll eliminate matching terms:
4e^2x - 6e^-x - 4e^2x + 6e^-x = 0
4e^2x - 6e^-x - 4e^2x + 6e^-x = 0 <=> f"(x) - f'(x) - 2f(x) = 0 q.e.d.