1. Expand the function \[ f(x)=\sin x, \qquad 0\leq x\leq \pi \] in terms of the eigenfunctions of the Sturm-Liouville problem: \[ \begin{cases} y''+\lambda y=0 \\ y(0)=0, \quad y(\pi)+y'(\pi)=0 \end{cases} \]
2. Consider the Sturm-Liouville problem \[ -\Big[ p(x) u' \Big]' = \lambda r(x) u , \qquad (a<x<b) \] with BCs \[ u(a)=u(b)=0 \] Assuming that \(p(x) >0\) and \(r(x)>0\) for all \(a\leq x\leq b\), show that the eigenvalues \(\lambda\) are all positive.
3. Use the method of eigenfunction expansions to solve the BVP problem: \(u''+ 4u = x^2\), \(u(0) = u(1) = 0\) after showing that it has a unique solution.
4. Consider nonhomogeneous boundary value problem: \[ u''+ u = f(x), \] with BCs \(u(2\pi)-u(0) = \alpha\), and \(u'(2\pi)-u'(0) = \beta\).
4a. Find the solutions \(v\) of the homogeneous adjoint problem \(L^*(v)= 0\), with adjoint BCs.
4b. Then impose \(\langle Lu, v\rangle = \langle f, v\rangle\) for all \(v\)’s to obtain the solvability condition.
4c. Assuming this condition to be satisfied, can function \(u\) be found by the method of eigenfunction expansions?