non-homogenous recurrence relation, with split boundary conditions

Question

I have non-homogenous recurrence relation:

$x_{t+1}=\alpha x_t+\beta x_{t-1}+\gamma$

with the following boundary conditions:

$x_2=\alpha x_1+\gamma$

$x_{T}=1/2x_{T-1} +1/2$

Anyone know how to solve this?

Thanks, Rob

Answer 1

The homogeneous solution is given by $x_H=c_1 r_1^t+c_2 r_2^t$ where $c_1,\;c_2$ are arbitrary constants and $r_i,\;\;i=1,2$, are the roots of the polynomial $r^2-\alpha r-\beta=0$. As implicitly suggested by BerkeleyChocolate, the particular solution is $x_P=\gamma/(1-\alpha-\beta)$. So, the general solution of the equation is $$x_G=c_1 r_1^t+c_2 r_2^t+\frac{\gamma}{1-\alpha-\beta}.$$ The boundary conditions give $$ c_1r_1(r_1-\alpha)+c_2r_2(r_2-\alpha)=(\alpha-1)x_P+\gamma, $$ $$ c_1r_1^{T-1}(r_1-1/2)+c_2r_2^{T-1}(r_2-1/2)=-x_p/2+1/2. $$ This gives a non homogeneous linear system for the constants $c_1$ and $c_2$.