Consider the differential equation

$$\frac{d x}{d t}+K x=0$$

where $K$ is a positive constant. By using the approximate finite-difference formula

$$\frac{d x_{n}}{d t}=\frac{x_{n+1}-x_{n-1}}{2 \delta t}$$

where $\delta t$ is a positive constant, and where $x_{n}$ denotes the function $x(t)$ evaluated at $t=n \delta t$ for integer $n$, convert the differential equation to a difference equation for $x_{n}$.

Solve both the differential equation and the difference equation for general initial conditions. Identify those solutions of the difference equation that agree with solutions of the differential equation over a finite interval $0 \leqslant t \leqslant T$ in the limit $\delta t \rightarrow 0$, and demonstrate the agreement. Demonstrate that the remaining solutions of the difference equation cannot agree with the solution of the differential equation in the same limit.

[You may use the fact that, for bounded $|u|, \quad \lim _{\epsilon \rightarrow 0}(1+\epsilon u)^{1 / \epsilon}=e^{u}$.]