In general, how does one solve an inhomogeneous second-order linear difference equation of the form

$$ a f(n+1) + b f(n) + c f(n-1) = d(n) $$

where $a, b, c$ are constants but $d(n)$ may also depend on $n$? For instance, consider as in here: $ 2 f(n+1) - 7 f(n) + 3 f(n-1) = 2 + 2^n $ with boundary conditions $q_0 = 1$ and $q_1 = 2$.

I understand that the general solution can be written as the sum of a particular solution to the inhomogeneous equation plus the general solution of the corresponding homogeneous equation, and I am able to find the latter, but how does one come up with a particular solution to the inhomogeneous equation in this case?

  • $\begingroup$ obtaining a particular solution can be really complicated. It depends a lot on the function type of $d (n)$. In the polynomial case it is usually direct. Also for some exponential cases. $\endgroup$ – Cesareo Dec 25 '18 at 9:02
  • $\begingroup$ @Cesareo Thanks; could you please elaborate how to do so for the polynomial and exponential cases such as $2 + 2^n$ in an answer? $\endgroup$ – p-value Dec 25 '18 at 21:10

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