Exact finite difference solution for constant second derivative ($y_{n-1}-2y_n+y_{n+1}=c{h}^2$)

On a uniform mesh $x_n=a+nh$, the centered second finite difference is exact for every quadratic polynomial. In particular, for $y^{\prime \prime }=c$ with endpoint boundary conditions, the unique discrete solution equals the exact quadratic solution at every mesh point. Subtracting consecutive first differences reduces the equation to $d_{n+1}-d_n=c{h}^2$, so $y_n=\frac{1}{2}c{h}^{2}n(n-1)+d_{1}n+y_0$. The second endpoint determines $d_1$. Uniqueness follows because any difference of two discrete solutions has zero second finite difference, hence is linear in the mesh index and vanishes at both endpoints. Exact nodal values in this example do not imply that a general finite difference method is exact for other ordinary differential equations.