I have been trying to solve the following functional equation
$$f(x+2)+af(x+1)+bf(x)=0$$ for all real values of $x$.
My guess and intuition leans towards $f(x)$ being some exponential function. I started out with $f(x)=\lambda a^x $ but ended up with a quadratic which seemed like a dead end.
Any hints will be helpful.
It does look similar to the general differential equation $${d^2y \over dx^2} + a {dy\over dx} + b=0$$ which also leads to a quadratic but I'm getting messy expressions for the functional equation.
$\mathrm{f(x) = k_1\alpha ^x + k_2\beta ^x}$ is the solution for the given functional equation, where $\mathrm{\alpha}$ and ${\beta}$ are the roots of the equation $\mathrm{x^2+ax+b=0}$.