yes sir, you are right. it is a system of 3 second order ODEs in three unknowns. but as i mentioned, mu(t) is a parameter just like a lagrange multiplier. i dont have any boundary condition on that variable. i am able to get the solution of the system without the presence of exponential terms in the system with the same BCs. dsolve,numeric treats is as a parameter and gives the solution.

but with the addition of the exponential terms in the equations, the initial newton iteration does not seem to converge

yes sir, you are right. it is a system of 3 second order ODEs in three unknowns. but as i mentioned, mu(t) is a parameter just like a lagrange multiplier. i dont have any boundary condition on that variable. i am able to get the solution of the system without the presence of exponential terms in the system with the same BCs. dsolve,numeric treats is as a parameter and gives the solution.

but with the addition of the exponential terms in the equations, the initial newton iteration does not seem to converge