@tomleslie epsilon is in my DTM worksheet but it's represented by zeta (ζ) in my numerical worksheet. sorry for the little mix up.

@acer Thank you so much. I actually need all the all the real roots

@tomleslie the worksheet will readily run as it is peacefully but by changing ε to 1 or a higher number, taking the limits to get values for A (f''(η)) and B (t'(η)) becomes problematic. thats where i'm having problems.

@Carl Love Thank you so much for your time. here it is. for DTM result DTM_result.mw and for the numerical result numerical_result.mw

@tomleslie I finally solved my problem. i really didnt have to go that far to get my result. the solution has been there all along. 

@tomleslie thanks alot. I just attached my codes above. I'm new to maple so I might not get somethings right but hopefully I will as time goes on. help will be greatly appreciated. And about the errors, I hope my code explains it but if i'm still wrong, corrections are gladly welcomed.

:) thanks in advance

@Carl Love I've tried all I can but I'm not getting the right answer. Could you please help with a complete DTM code for the problem to see my mistakes. :(

@Preben Alsholm  Thanks for the code. It sure works but if I can't solve it using Differential Transform Method then I'm still stuck. :(

Thanks very much. The values of A and B are f''(0) and θ'(0) since Differential Transforms method is an iterative procedure for getting the analytic solutions of the problem, A and B appears in F(2) = A/2; and Θ(1) = B which are transformed functions using DTM. Similar transformation already gave F(0) = 0; F(1) = 0; and Θ(0) = 1. Since I have these, I can then use them to get F(k) from the equations where k>2 and Θ(k) where k>1. The series solution is then obtained. The pade approximant must be applied to the series solution and by using asymptotic boundary condition (f'(∞)→1, θ(∞)→0) we can obtain A and B@Preben Alsholm 

Thank you so much for your time. Here's the real problem

f'''(η) + 3f(η)f''(η) - 2[f'(η)]² + θ(η) - m*f'(η) = 0

θ''(η) + 3*Pr*f(η)θ'(η) + s*θ(η) = 0

Boundary conditions are:

at η=0: f(η)=f'(η)=0; θ(η)=1;

as η→∞ f'(η)=1; θ(η)=0;

Where m = magnetic parameter (in this case taken as 2)

S = shrinking parameter (in this case taken as 1)

Pr = taken as 1 too

Thanks you! Thank you! :) @Carl Love 

Thank you so so much! It worked just fine. But my last problem is that my results for A and B using DTM aren't agreeing with the shooting method solution. Shooting method gives A and B as as 0.742408087440 and 0.943866213084 respectively. I don't know what I'm doing wrong but everything seems correct to me. @Carl Love