I have a battle with the following direct partial integration and separation of variables toffee: I have to solve, [itex]u(x,y)=\sum_{n=1}^{∞}A_n sin\lambda x sinh \lambda (b-y)[/itex] If there were no boundary or initial conditions given, do I assume that λ is [itex]\frac{n\pi}{L}[/itex] and do I then solve [itex]A_n[/itex]? If I am going in the wrong direction here, please point me in the right direction... thanks!
As far as I know, there is no way to solve this further without boundary conditions. You need a condition of the type [itex]u(x_0,y)=g(y)[/itex] or [itex]u(x,y_0)=g(x)[/itex]. By evaluating the equation with the boundary condition, you can use Fourier series to find the coefficient [itex]A_n[/itex].
Meldraft: Thank you for your contribution. I agree and am also expecting boundary conditions... I want to carry on using the Fourier approach to solve An but due to the lack of boundary conditions, I find it impossible unless I miss something. I am not sure if the Eigen separation constant (λ) may lead to solving the 'boundary conditions" in some degree based on the arrangement of the equation (hyperbolic)... I am stumped...
I am not sure that I understand what your question is. You have a general solution. If you had boundary conditions you could define the constant, but as it is, there is really nothing more you can do. Are you trying to solve it without applying a boundary condition? If so, PDE theory suggests that there are infinite possible solutions, so you cannot possibly get a unique value for your constant, without applying a boundary condition!