Exercise 1.3.13

Odd size: The five centered difference equations lead to $b_1+b_3+b_5=0$.

$\begin{array}{ccccc}\hfill x_2& =b_1\hfill & \hfill & \text{ Add equations }1,3,5\hfill & \hfill \\ \hfill x_3-x_1& =b_2\hfill & \hfill & \text{ The left side of the sum is zero }\hfill \\ \hfill x_4-x_2& =b_3\hfill & \hfill & \text{ The right side is }{b}_1+b_3+b_5\hfill \\ \hfill x_5-x_3& =b_4\hfill & \hfill & \text{ There cannot be a solution unless }{b}_1+b_3+b_5=0.\hfill \\ \hfill -x_4& =b_5\hfill & \hfill & \hfill \end{array}$