Consider the problem: [frac{partial^{2} G}{partial x^{2}}=deltaleft(x-x_{0} ight), quad frac{partial G}{partial x}left(0, x_{0} ight)=0, quad Gleft(pi, x_{0} ight)=0]

Consider the problem:

\[\frac{\partial^{2} G}{\partial x^{2}}=\delta\left(x-x_{0}\right), \quad \frac{\partial G}{\partial x}\left(0, x_{0}\right)=0, \quad G\left(\pi, x_{0}\right)=0\]

a. Solve by direct integration.

b. Compare this result to the Green's function in part b of Problem 31.

c. Verify that \(G\) is symmetric in its arguments.