Extension: Solving y'' + ay' + by = 0

Which of the following twice-differentiable real functions $f \colon \mathbb{R} \to \mathbb{R}$ satisfy the "difference of squares" functional equation: $f(x)^2 - f(y)^2 = f(x + y) f(x - y)$ (for all reals $x,y$).

Source: Putnam 1963