I unsuccessfully attempted to separate the differential equation problem from the algebraic problem.

I had a different idea that you might see if you could use. Differentiate both sides of your Eq111. That gives you the needed differential equation for H[z]. Since that adds another differential equation, you may or may not need another initial condition to uniquely reconstruct H[z].

When I took a moment and tried to do this I ended up with multiple warnings about 1/0. I didn't explore whether it might be possible to avoid those by getting rid of denominators or if this was because of something else. But if you can track down the cause of those complaints and resolve them then this might get you where you need to go. Or at least you might learn a little more about your system.

Hello Bill, thank you so much for your answer.

About the '' yes, you are right, I noticed that a few minutes after posting this thread. About the mathematical problems, we have H[z] which does not have any derivative (there are no H[z]' ) and psi[z] which is a second derivative. The thing is that psi[z] depends on H[z] and vice-versa. About the boundary conditions, one of them (psi'[0]) depends on the result of NDSolve, which is psi[z].. So I really don't know how to solve complex system... If you have any idea that might help I'll really appreciate!

Thanks! :)