= Solution
\b[The printed formula is not a consistent spatial discretization of the stated PDE.] The original PDF genuinely has $1/\Delta x$, rather than $1/(\Delta x)^2$. Writing $h=\Delta x$, the <Taylor expansion> gives
$$
h^{-1}\bigl(u(x-h)-2u(x)+u(x+h)\bigr)=h\,u_{xx}(x)+O(h^3),
$$
so it approximates a diffusion coefficient tending to zero. Moreover, $m=1,\ldots,M$ and $h=1/(M+1)$ cover $(0,1)$, whereas the given <Dirichlet boundary conditions> belong to $(-1,1)$. No value at the interior location $x=0$ was prescribed. These are actual source defects, not repairs justified by the TeX conversion.
A conditional analysis of the literal <ordinary differential equation> system is still possible. If it is closed by imposing $u_0=u_{M+1}=0$ on the displayed grid, the negative <Dirichlet discrete Laplacian> has positive frequencies squared
$$
\kappa_j(h)=\frac4h\sin^2\frac{j\pi}{2(M+1)},\qquad j=1,\ldots,M,
$$
and each <normal mode> obeys $q_j''=(\alpha-\kappa_j)q_j$. For a fixed mesh, every solution is bounded for all time exactly when
$$
\boxed{\alpha<\kappa_1(h)=\frac4h\sin^2\frac{\pi h}{2}}.
$$
Equality produces a linearly growing <normal mode> for nonzero modal initial velocity; above the threshold there is an exponentially growing <normal mode>. Since $\kappa_1(h)\sim\pi^2h\to0$, any $0<\alpha<\pi^2/4$ gives bounded continuous solutions but unbounded literal discrete solutions on sufficiently fine meshes. For example, $\alpha=1$, $M=99$ gives $\kappa_1<1$, and a growing first sine <normal mode>. At $\alpha=0$, every fixed mesh is bounded for all time, but unit first-mode initial velocity has maximal displacement $1/\sqrt{\kappa_1(h)}$, which diverges on mesh refinement. For uniform-in-mesh, all-time displacement bounds for arbitrary bounded displacement/velocity data in the mesh-weighted <L2 norm>, the literal closed system therefore requires $\alpha<0$.
This differs from finite-time <stability of a numerical method>, governed here by <displacement stability of a symmetric semidiscrete wave equation>. For the conditional closure above, <diagonalization of a matrix> gives, with $\alpha_+=\max(\alpha,0)$,
$$
\|U(t)\|_h\leq e^{\sqrt{\alpha_+}t}
\bigl(\|U(0)\|_h+t\|U'(0)\|_h\bigr),\qquad
\|U\|_h^2=h\sum_m|U_m|^2.
$$
Oscillatory modes use $|\sin(\omega t)/\omega|\leq t$, including its value $t$ at $\omega=0$; growing modes use $\sinh(\beta t)/\beta\leq t e^{\beta t}$. Thus the displacement map is mesh-uniformly bounded on each fixed finite time interval for every fixed real $\alpha$. This is a legitimate finite-time displacement <stability> estimate, but neither <consistency of a numerical method> nor the all-time conclusion of part (i). It does not assert a velocity bound in the unweighted displacement norm.
If the intended method instead uses $h^{-2}$ and the full interval, take interior indices $m=-M,\ldots,M$, $h=1/(M+1)$, with zeros at $m=\pm(M+1)$. Its first positive discrete <eigenvalue> is
$$
\lambda_{1,h}=\frac4{h^2}\sin^2\frac{\pi h}{4}<\frac{\pi^2}{4},
\qquad \lambda_{1,h}\longrightarrow\frac{\pi^2}{4}.
$$
The corrected, fixed-mesh all-time condition is $\alpha<\lambda_{1,h}$, with the same strictness at equality. For every fixed $\alpha<\pi^2/4$, this holds on all sufficiently fine meshes. This corrected interpretation is stated separately rather than silently replacing the PDF.
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