I am a bit confused about the entire topic of the adjoint and am not totally sure how to apply the definition of it to a given differential equation in order to obtain the adjoint equation. I have perused several examples but have not found one that fits well with the form of my equation shown here adjoint problem
Specifically, I would appreciate an explanation of the process to obtain the adjoint for this equation (I believe it is mostly an application of integration by parts) as well as the application of the current boundary conditions given and whether they change with the adjoint function.