[Fahraynk]

I have a system which I solved with newtons iteration, and I have to add an additional unknown to the system. The system is the following :

I broke it up into a system of 3 equations:




m is a constant; I set manually set m.
Unknowns are f, Fp, Fpp.
The equations are approximated with taylor's approximations, and G(x)=0 is the sum of these 3 equations at 1000 grid points + 3 initial conditions.
Currently, , and I use newtons iteration to solve for X such that G(X)=0

This problem I am trying to solve is this, I am given this information :





I am told that I have to take a given H, and use newtons iteration to find m, the constant which I was setting manually before.

I want to know, if I change X to and then add to G(x), ,will this be enough for newtons method to solve for m? Or do I have to do something else?