Discussion
For the solution of \(\nabla^{2} u = 0\), the domain and boundary conditions are shown below.
Which of the following statements is TRUE?
Answer: C
The true statement is that the solution cannot be obtained using separation of variables because not all the boundaries are along constant coordinate lines.
Separation of variables in Cartesian coordinates needs a domain whose sides are lines of constant x or constant y, so that each boundary condition can be imposed on one factor of the product solution. Here the left side is a slanted line through the origin, which is neither, so the method fails for this domain.
Laplace's equation itself is separable, so the first option is wrong. Non-zero boundary values are handled by superposing several separated solutions, so the second is wrong. And the slanted edge means the fourth option is wrong as posed.
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