As in Figure 9.29, consider the rectangular region D defined in the xy-plane by and . Let , and choose the coefficients so that takes on prescribed boundary values at the corners; recall Example 1. These requirements lead to the system of linear equations
where the represent the boundary corner values. By explicitly solving the system, show that this system has a unique solution for any positive values of a and b and for any choice .
From the system of equations, we immediately have . Using the equation we have
Next we use and to derive
Finally we use , , , and to derive