I am an undergraduate student and I want to know how to solve this hydrodynamic stability problem. I tried but all my efforts have failed. Your suggestions will help me a lot. The equations are f''- a^2 f + a R h=0; h''- a^2 h + a f=0; And the conditions at the plates are z=0: f=0,h=0 and z=1: f=0, h=0. In my first attempt, I am fixing a=3.1472 and then find R as an eigenvalue and print the eigenfunctions f and h. To do this I worte the following code by introducing one more equation for R i.e R'[z]=0 with an artificial condition R[0]=n. The following code represents this.

Block[{a = 3.1479}, 
sol = ParametricNDSolve[{f''[z] - a^2 f[z] + a R[z] h[z] == 0, 
h''[z] - a^2 h[z] + a f[z] == 0, R'[z] == 0, f[0] == f[1] == 0, 
h[0] == h[1] == 0, R[0] == n}, {f[z], h[z], R[z]}, {z, 0, 1}, n]]

I would be thankful if you can help me to solve this problem.