# Solution to recurrence relation from integro-differential dynamical system?

Consider the integro-differential equation

$$\begin{equation} \kappa\ddot x+\dot x=2\int_0^t J_1(x_t-x_s)e^{-\epsilon(t-s)}ds.\tag{1} \end{equation}$$

such that $$\kappa,\epsilon\in\mathbb{R}$$, $$t\in\mathbb{R}^+$$, and $$J_1$$ is first order the Bessel function of first kind. Letting $$\begin{equation} \mathcal{S}:=\sum_{i=0}^n J_1\left(x_{n\Delta t}-x_{i\Delta t}\right)e^{-\epsilon\left(n\Delta t-i\Delta t\right)},\tag{2} \end{equation}$$

Equation $$(1)$$ may be expressed as the system $$\begin{equation} x_{\Delta t(n+1)}=\lim_{\Delta t\rightarrow 0}\frac{2\Delta t^2 \mathcal{S} + x_{n\Delta t}\left(2\kappa+1\right)-\kappa x_{\Delta t(n-1)}}{\kappa+1},\tag{3} \end{equation}$$ such that $$x_0=0$$ and $$x_1\in\mathbb{R}$$. This is particularly useful as a numerical method when solving the first equation. However, can a recurrence relation be derived from the above? Setting $$\Delta t=1$$ and letting $$\mathcal{S}$$ be constant simply produces $$x_{n}=K_1\left(\frac{\kappa}{1+\kappa}\right)^n+2\mathcal{S}\left(n-1-\kappa\right)+K_2,$$ for constants $$K_1,K_2$$. Does a more general relation exist for the function $$\mathcal{S}$$ and $$0\leq\Delta t<1$$?