First, rename the constants appropriately so that we get $d_t = d_{t-1} + A v_{t-1} \\ v_t = x_t + B v_{t-1} + Cd_{t-1}$
Then use the second equation to substitute $d_t$ and $d_{t-1}$ in the first one: $ (v_{t+1}-x_{t+1}-Bv_t)/C = (v_t - x_t -Bv_{t-1})/C+Av_{t-1} \\ \Leftrightarrow v_{t+1} -(B+1)v_t +(-AC+B)v_{t-1}=(x_{t+1}-x_t) $
This is an inhomogeneous second order difference equation. There are standard methods to solve this. You first find the solutions for the homogeneous case (i.e. $x_{t+1}-x_t=0$) and then you find one particular solution to the inhomogeneous case. This will obviously depend on the nature of the $x_t$, and there may not be a closed form. Then you can substitute back to get $d_t$. So if your input is real data you may have to put up with numerical iterative solutions for $d_t$.