= Solution
With $h=\delta t>0$, the <central finite difference> converts the decay equation to
$$
\boxed{x_{n+1}+2Khx_n-x_{n-1}=0.}
$$
The continuous <ordinary differential equation> has the <general solution> $x(t)=Ce^{-Kt}$. The discrete <linear recurrence> is second order and requires two initial values, such as $x_0,x_1$. Trying $x_n=\rho^n$ gives the <characteristic equation of a linear recurrence> $\rho^2+2Kh\rho-1=0$. Its roots are
$$
\rho_+=-Kh+\sqrt{1+K^2h^2},\qquad
\rho_-=-Kh-\sqrt{1+K^2h^2},
$$
so
$$
\boxed{x_n=A\rho_+^n+B\rho_-^n,\quad
A=\frac{x_1-\rho_-x_0}{\rho_+-\rho_-},\quad
B=\frac{\rho_+x_0-x_1}{\rho_+-\rho_-}.}
$$
There are no repeated roots, and the same formula extends to negative integer $n$ if required.
To demonstrate the continuum limit, put $a_h=\operatorname{arsinh}(Kh)$. Then $\rho_+=e^{-a_h}$ and $\rho_-=-e^{a_h}$, with $a_h/h\to K$. For $t_n=nh\in[0,T]$,
$$
\rho_+^n=e^{-(a_h/h)t_n}\longrightarrow e^{-Kt_n}
$$
uniformly, since $0\le t_n\le T$. More precisely $a_h/h=K-K^3h^2/6+O(h^4)$, giving an $O(h^2)$ error on a fixed interval. Thus $B=0$, $A=C$ selects the physical discrete mode and corresponds to $x_1=\rho_+x_0$.
The other root is a <parasitic amplification root>:
$$
B\rho_-^n=B(-1)^n e^{(a_h/h)t_n}.
$$
For fixed $B\ne0$, even and odd mesh points approach values differing by $2Be^{Kt}$. Equivalently successive mesh values have a nonvanishing alternating jump despite their time separation tending to zero. This mode cannot converge uniformly to a continuous ODE solution; its limiting envelope also grows rather than decays.
For step-dependent initial data the precise result is the <centered two-step discretization of exponential decay> criterion:
$$
\boxed{x_n\to Ce^{-Kt_n}\text{ uniformly on }[0,T]
\quad\Longleftrightarrow\quad A_h\to C\text{ and }B_h\to0.}
$$
Sufficiency follows from the two uniformly bounded exponential envelopes on a fixed interval. Necessity follows from convergence at the adjacent points $0,h$: $x_0\to C$, $x_1-x_0\to0$, and the coefficient formulas force $B_h\to0$, $A_h\to C$. In particular a nonzero but vanishing starting error is not excluded. Starting with the exact value $x_1=Ce^{-Kh}$ gives $B_h=CK^3h^3/12+O(h^4)$ and still converges on each fixed interval. The assertion about “remaining” solutions therefore concerns a fixed nonzero parasitic amplitude, not every nonzero step-dependent amplitude. The method is nevertheless not damping-stable for arbitrarily long times because $|\rho_-|>1$.
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