(a) Determine the Green's function $G(x, \xi)$ for the operator $\frac{d^{2}}{d x^{2}}+k^{2}$ on $[0, \pi]$ with Dirichlet boundary conditions by solving the boundary value problem

$$\frac{d^{2} G}{d x^{2}}+k^{2} G=\delta(x-\xi), \quad G(0)=0, G(\pi)=0$$

when $k$ is not an integer.

(b) Use the method of Green's functions to solve the boundary value problem

$$\frac{d^{2} y}{d x^{2}}+k^{2} y=f(x), \quad y(0)=a, y(\pi)=b$$

when $k$ is not an integer.