thank you very much Preben! finally i got the result from your suggestions!tears...

may your life be filled with blessings!

i already put diff(G(eta),eta)=g(eta) with G(0)=0 in the system as follows

> FNS := {G(eta), f(eta), f1(eta), f2(eta), g(eta), g1(eta)};
> ODE := {diff(g1(eta), eta)+f(eta)*g1(eta)-f1(eta)*g(eta) = 0, diff(f2(eta), eta)+f(eta)*f2(eta)+1-f1(eta)*f1(eta) = 0, diff(G(eta), eta) = g(eta), diff(f(eta), eta) = f1(eta), diff(f1(eta), eta) = f2(eta), diff(g(eta), eta) = g1(eta)};
            
> IC := {G(0) = 0, f(0) = fo, f1(0) = -0.5, f2(0) = alpha, g(0) = 1, g1(0) = beta};
 
How can i get the numeric values for the integration for ∫g(η)dη (integrate from 0 to 10)?

@Preben Alsholm 

thank you very much for your comments.

To answer your questions, i used the shooting method to solve the system, lambda = -0.5 and in place of infinity i put 10. i want to solve for eta = 0 until 10.

from the shooting method, i will get the following result where alpha=f ''(0) and beta = g '(0)

> S := shoot(ODE, IC, BC, FNS, [alpha = 2, beta = 0]);
newton: Step #  1
newton: Parameter values :  alpha = 2 beta = 0
newton: Step #  2
newton: Parameter values :  alpha = 1.81139757054676087 beta = -.131600480171221584
newton: Step #  3
newton: Parameter values :  alpha = 1.67969492778382000 beta = -.230621431328498994
newton: Step #  4
newton: Parameter values :  alpha = 1.59253568378099962 beta = -.307651731170162146
newton: Step #  5
newton: Parameter values :  alpha = 1.53882738922096118 beta = -.371769405101317951
newton: Step #  6
newton: Parameter values :  alpha = 1.50968203001711854 beta = -.429501899609736948
newton: Step #  7
newton: Parameter values :  alpha = 1.49796574416486682 beta = -.478096148292238066
newton: Step #  8
newton: Parameter values :  alpha = 1.49574971779388566 beta = -.499668285226424590
newton: Step #  9
newton: Parameter values :  alpha = 1.49566990254340860 beta = -.501442526935696131
newton: Step #  10
newton: Parameter values :  alpha = 1.49566979992870364 beta = -.501447611044229524

but i am not sure how to get g from the result. i can just plot g(eta) as follows

> q := odeplot(S, [eta, g(eta)], 0 .. 10);
> display(q);