Consider the following couples systems of equations
f′′− k^2 f=kRa g′′− k^2 g=kf , where f and g are functions of z. The boundary conditions for the problem are f=0 and g=0 at z={0,1}. Here we have two parameters Ra (Rayleigh number) and k (wave number).
Now, I want to find the minimum of Ra as k varies. How do we find this? Also, I want to find and plot the eigenfunctions f and g in the interval. I would like to do this using Mathematica.