Consider the boundary value problem
$$
-\frac{d}{d x}\left(a(x) \frac{d u}{d x}\right)=f(x), \quad u(0)=u(1)=0
$$
where $a(x)>\delta \geq 0$ is a bounded differentiable function in $[0,1]$. We assume that, although $a(x)$ is available, an expressi on for its derivative, $\frac{d a}{d x}$, is not available.
(a) Using finite differences and an equally spaced gird in $[0,1], x_l=h l, l=0, \cdots, n$ and $h=1 / n$, we discretize the ODE to obtain a linear system of equations, yielding an $O\left(h^2\right)$ approximation of the ODE. After the application of the boundary conditions, the resulting coefficient matrix of the li near system is an $(n-1) \times(n-1)$ tridiagonal matrix.
Provide a derivation and write down the resulting linear syste $\mathrm{m}$ (by giving the expressions of the elements).
(b) Utilizing all the information provided, find a disc in $\mathbb{C}$, the smaller the better, that is guaranteed to contain all the eigenv alues of the linear system constructed in part (a).