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Put $h=1/(N+1)$ and $x_i=ih$. The interior piecewise-linear hat functions areThey are a basis of the continuous piecewise-affine functions vanishing at both endpoints. The support of a function for each hat only overlaps those of neighbouring hats, so the stiffness matrix is a tridiagonal matrix. Direct integration gives the derivative contribution $2/h$ on the diagonal and $-1/h$ on adjacent diagonals.

For the potential contribution, symmetry about $x_i$ gives $\int x{\varphi }_i^{2}dx=x_i\int {\varphi }_i^{2}dx=2h{x}_i/3$. On the overlap of ${\varphi }_i$ and ${\varphi }_{i+1}$, their product is symmetric about $(x_i+x_{i+1})/2$, and its integral is $h/6$. This is the affine-weighted hat mass matrix calculation. Consequently,and all other entries vanish. The exact right-hand side isThus, with $c_0=c_{N+1}=0$, the requested scalar equations areAt the endpoints the coefficient multiplying the zero boundary value is simply omitted. Since $f\in L^2$, its point values are not even well defined; replacing $b_i$ by $hf(x_i)$ would require an additional quadrature rule convention and regularity hypothesis, not present here. The exact Ritz method uses the displayed integrals. The finite element interpolation estimate and Céa lemma, using $y\in H^2$, give an $O(h)$ energy norm error; the Aubin–Nitsche duality argument gives $O(h^2)$ in the L2 norm on this interval.