# How to solve a non-homogeneous second-order linear difference equation with both a forward and a backward difference?

Quite a long title for this: I'm looking for the general solution of the following difference equation: $$ax_{t+1} -bx_t + x_{t-1} = c + u_t$$ where $a,b,c$ are real constants and $u_t$ is a bounded stochastic disturbance (e.g. a uniformly distributed random variable between -1 and 1 and the $u_t$ are iid).

This is the class of difference equations, that Woodford (2003, Interest and prices, chapter 7) solves as the law of motion of Lagrange multipliers $x$. Unfortunately, he just states "Well, folks, this is the result" but does not bother explaining or even stating his approach.

Now, I'm pretty sure this problem can be tackled with the usual approach for second-order non-homogeneous difference equations with constant coefficients, i.e. the solution should be looking like this (the condition for two distinct real roots of the CE is fulfilled): $$x_t = Am_1^t + Bm_2^t + x^*$$

Is this true or am I missing something?

• Yes, that is right. But I've got no clue on how to handle the forcing $u_t$. – vonbrand Aug 18 '15 at 17:23