0

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.

statsisfun_sid
  • 5
  • The problem is simple for integer $x$ – Claude Leibovici May 29 '21 at 08:39
  • @ClaudeLeibovici I wish to solve $f(x)$ for all real values of $x$ – statsisfun_sid May 29 '21 at 10:03
  • Why would the quadratic seem like a dead end? It can give you some solutions. – Sil May 29 '21 at 11:37
  • 1
    Essentially, the result for the integers is all you can get. More precisely, for any $ g : [ 0 , 2 ) \to \mathbb R $, you can extend $ g $ to an $ f : \mathbb R \to \mathbb R $ satisfying the desired equation; just treat every $ x \in [ 0 , 1 ) $ separately, and find the value of $ f ( x + n ) $ for all $ n \in \mathbb Z $. Even further assumptions on $ f $ like continuity/smoothness doesn't make the solution set very small (you'd only need $ g $ to be continuous/smooth with the additional assumption that its limit or the limits of its derivatives near $ 2 $ equal to the value at $ 0 $). – Mohsen Shahriari May 29 '21 at 22:26
  • 1
    The situation is different if you restrict the case more than that, and for example require $ f $ to be real analytic. in that case, the only solutions are those that are (linear) combinations of functions of the form you've already mentioned. – Mohsen Shahriari May 29 '21 at 22:27

1 Answers1

0

$\mathrm{f(x) = k_1\alpha ^x + k_2\beta ^x}$ is the solution for the given functional equation, where $\mathrm{\alpha}$ and ${\beta}$ are the roots of the equation $\mathrm{x^2+ax+b=0}$.

Random Person
  • 1
  • 2
    As it’s currently written, your answer is unclear. Please [edit] to add additional details that will help others understand how this addresses the question asked. You can find more information on how to write good answers in the help center. – Community Aug 05 '22 at 16:38