>In many physical and engineering problems invloving
>differential equations in a function F(x,y,z), we search the
>solutions by asuming that F(x,y,z) is of the form
>
(*) F(x,y,z) = X(x)Y(y)Z(z).
>
>But what I haven't read yet is the proof that all posible
>solutions are of this form. Does anyone know the proof?
>

There is none. Call a solution in the "separable" form (*) primitive. They often exist with some or all X, Y, Z from specific function families: trigonometric, exponential, and others. More general solutions are looked up as a combination - finite or infinite - of primitive solutions.