Particular solution to nonhomogeneous difference equation.

Question

I'm working on one of the problems in Edelstein-Keshet's book 'Mathematical models in Biology'. The problem in question asks me to show that for some constant $K$, $X_{n}=K$ is a solution to the equation $aX_{n+2}+bX_{n+1}+cX_{n}=d$, with $d\neq 0$. The excercise also states that for that to be true, 1 must not be a solution of the characteristic equation, and I just can't see why that would be a problem. Unless I'm misinterpreting the problem the only requirement should be that $a+b+c\neq 0$.

Answer 1

It took me way too long to realize this. $a+b+c=0$ is equivalent to $x=1$ being a solution to $ax^{2}+bx+c=0$.