The Main Result

$ \def \EN {\quad \mbox{and} \quad} $ Consider a one-dimensional uniform mesh with grid-spacing $dx$. Neighbouring grid-points in the mesh are coupled by coefficients $a$ in forward direction and by coefficients $b$ in backward direction. Setting up linear equations for such a grid gives rise to a tri-diagonal matrix, which can be (re-)normalized to obtain $1$'s on the main diagonal: $$ \left[ \begin{array}{ccccccc} . & . & . & & & & \\ & -b & 1 & -a & & & \\ & & -b & 1 & -a & & \\ & & & -b & 1 & -a & \\ & & & & . & . & . \end{array} \right] $$ The system of equations can be solved by employing a Newton-Raphson MultiGrid method. By employing the requirement that Properties of the tri-diagonal system should be Persistent on any coarsened or refined grid, we find that the Rule of Positive Coefficients is universally valid. And the discriminant of the equations system must be positive: $$ a \ge 0 \EN b \ge 0 \EN 1 - 4.a.b \ge 0 $$ In the limiting case of a immensely fine grid, the system of equations becomes associated with a linear ODE of second order, called the Governing Equation: $$ A \frac{d^2T}{dx^2} + B \frac{dT}{dx} + C.T = 0 $$ Where $x=$ coordinate, $A,B,C=$ constants and $T(x)=$ solution.
It is conjectured that the discriminant of the characteristic equation of the ODE must be positive (or zero). Having investigated all special cases, this turns out to be the only condition: $$ B^2 - 4.A.C \ge 0 $$ Hence, oscillatory solutions can never be described by the governing ODE of a persistent tri-diagonal system of equations.
The coefficients $a$ and $b$ can be expressed in the coefficients $A,B,C$ of the governing ODE and the grid-spacing $dx$. The solution of the tri-diagonal system is nothing else but a sampling on the grid of the Analytical Solution belonging to the governing ODE.