I think you are trying to solve three coupled non-linear differential equations, i.e. six differential non-linear equations simultaneously, you can either take as examples of Newton's method for solving system of non-linear equations or Runga-kutta (RK-4) methods. Further you can see my one paper "Dynamic behavior of Underwater towed cable in linear profile", where I used Newton's method to solve six non-linear differential equations simultaneously.

I think you are trying to solve three coupled non-linear differential equations, i.e. six differential non-linear equations simultaneously, you can either take as examples of Newton's method for solving system of non-linear equations or Runga-kutta (RK-4) methods. Further you can see my one paper "Dynamic behavior of Underwater towed cable in linear profile", where I used Newton's method to solve six non-linear differential equations simultaneously.

Suppose the system of ODE is dY/dt = [ f1(t,Y), f2(t,Y), ….., fn(t,Y)]t where Y is the unknown column vector [y1(t), y2(t), …., yn(t)] t . The initial vector Y (0 ) is given. If I apply RK4 method, then with the help of given right hand side column, compute the columns K1, K2, K3, K4 . Now new solution Y(t + dt) = Y(t) +(K1 + 2*K2 + 2*K3 + K4)/6 . In this way further solution can be obtained.

Thanks for your guidance. I will try the methods and if there would be any questions I would post it here :)

Fine.

So prof. R.Mittal you mean that as longs as I include all f1(t, Y) till fn(t,Y) in the loop, it would solve the equations simultaneously?

The point is f1 is somehow used in f2 and f3 while f3 has also f6 and f1 in it ( or something the same) so this is the main reason that these equations should be solved simultaneously.