Assignment:
Consider the Helmholtz partial differential equation:
uxx + uyy +(k^2)(u) =0
Where u(x,y) is a function of two variables, and k is a positive constant.
a) By putting u(x,y)=f(x)g(y), derive ordinary differential equations for f and g.
b) Suppose the boundary conditions are that u(x,y) vanishes on the lines x=0 ,x=3, y=0, and y=2. Derive the corresponding boundary conditions for f and g.
c) Given k^2, show that only certain values of the separation constant lead to non-trivial solutions for both f and g.
d) Find the non-trivial solutions of the differential equations for u(x,y) for the given boundary conditions.
e) For k^2 =2(pi)^2, obtain the general form of the solution u(x,y) of the partial differential equation compatible with the boundary conditions.
Provide complete and step by step solution for the question and show calculations and use formulas.