Consider the following boundary value problems. Determine the eigenvalues \(\lambda\) and eigenfunctions \(y(x)\) for each problem.

a. \(y^{\prime \prime}+\lambda y=0, \quad y(0)=0, \quad y^{\prime}(1)=0\).

b. \(y^{\prime \prime}-\lambda y=0, \quad y(-\pi)=0, \quad y^{\prime}(\pi)=0\).

c. \(x^{2} y^{\prime \prime}+x y^{\prime}+\lambda y=0, \quad y(1)=0, \quad y(2)=0\).

d. \(\left(x^{2} y^{\prime}\right)^{\prime}+\lambda y=0, \quad y(1)=0, \quad y^{\prime}(e)=0\).