Assumption of solutions in partial differential equations In Griffiths' Introduction to Electrodynamics, in all the problems regarding calculation of potential within grounded metal pipes and plates, while solving Laplace's Equation, the solution has been assumed to be a product of functions of $x$ and $y$. It has been highlighted in the figure below. Griffiths himself has written that the assumption is totally absurd.

My question is:
Why do we make such an assumption which only gives us special solution? Why don't we go for a general solution using some standard procedure of solving the pde?Is there some kind of advantage of using this type of an assumption?
 A: For linear, partial differential equations (wave equation, Fourier's heat equation, Schrödinger equation, diffusion equation (Fick's second law), convective diffusion equation and quite a few others) the method of separation of variables seems to always work (and can be adapted even for equations with source terms, i.e. non-homogenous PDEs).
If a function $u(x_1,x_2,x_3,...,x_n)$ is sought then the Ansatz is a function:
$$u(x_1,x_2,x_3,...,x_n)=X_1(x_1)X_2(x_2)X_3(x_3)...X_n(x_n)$$
Inserting the Ansatz in the original PDE and minimal reworking then allows separation of variables in the form of a number of ODEs:
$$f_1[X_1(x_1)]=f_2[X_2(x_2)]+f_3[X_3(x_3)]+...+f_n[X_n(x_n)]$$
Introducing a separation constant like $-m^2$ then gives:
$$f_1[X_1(x_1)]=f_2[X_2(x_2)]+f_3[X_3(x_3)]+...+f_n[X_n(x_n)]=-m^2$$
We then solve $f_1[X_1(x_1)]=-m^2$ using relevant boundary conditions. Once $-m^2$ is determined we can also write:
$$f_2[X_2(x_2)]=-m^2-f_3[X_3(x_3)]-...-f_n[X_n(x_n)]=-o^2$$
So we can solve:
$$f_2[X_2(x_2)]=-o^2$$
The process is repeated for all variables.
A nice example is my answer to this SE question.
Another step-by-step example: wave equation for an elastic string.
Note: the sign of the separation constant $-m^2$ has to be evaluated: it can be zero, negative or positive.

Is there some kind of advantage of using this type of an assumption?

The advantage is that it's generally simple and seems to always work. The obtained solution can of course be easily verified by re-inserting into the original PDE.
