Hello, Scholars, I'm trying to study the linear stability of stratified flows using Taylor-Goldstein equation and I have a linear second order Boundary Value Problem with eigenvalues 'c' and variable coefficients. I have been thinking of using shooting method but I need help on how to approach the problem. This problem has no analytical solution at the moment but I think Numerical approach can give some insight with real eigenvalues.

 Below is the equation with vertical stratification and 'w' is the vertical velocity of the wave

w''(z)+[(N_0^2*(exp(-2*z/h))/(U(z)-c)^2-(U''(z)/(U(z)-c)-k^2]w(z)=0

with boundary conditions

 w(z=0)=0, w(z=100)=0 where,

 where,

 c= phase velocity of the wave or the eigenvalues

U(z)=log(1+z/400), k=2*(pi)/lambda, lambda= 50meters

Many thanks

Joseph

 I will highly appreciate your responses.