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    Solutions to the Helmholtz Equation

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    Please help with the following problem. Provide step by step calculations for each problem.

    Consider the Helmholtz partial differential equation:
    u subscript (xx) + u subscript (yy) +(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.

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    Solution Summary

    We solve various cases of the Helmholtz equation by separating variables and performing Fourier expansions where necessary.