@dharr I see. This looks much better than the previous ones. Thanks for the code!

Thanks @dharr. This is quite nice, particularly the Taylor expansion at the end. I think this will resolve many issues in other ODEs that I want to solve with Maple.

But I still feel this is an additional and perhaps unncessary work that I should do, that could have been avoided if Maple directly gave me something like Mathematica. Mathematica is a huge pain, even impossible, when it comes to functional derivatives and field theory Poisson brackets and Maple is great for that, and that is why I want to run everything in Maple. But it seems that in these kinds of DEs, Mathematica works way better, and for more complicated situations where the dependence of c__1 and c__2 on beta__c is more complicated than a simple polynomial, I need to solve the DEs in Mathematica and copy the solutions form Mathematica to Maple, and then proceed.    

Thanks @dharr. Nice method! This is almost what I want, and I say almost because the hint that c__2 depends on beta__c comes from the solution that Mathematica gives us. I want Maple to give me an explicit solution with beta__c in both c and p__c  in other cases that I don't know Mathematica's solution, and furthermore these solutions do not vanish for beta__c=0.

In any case, based on your method, I wrote a new code that makes c and p__c themselves beta__c dependent and then I fix f_1(beta__c) and f_2(beta__c) based on what I know from Mathematica.
If you or anyone else comes up with a method that does not need all these manupilations and prior knowlege from Mathematica, and yields the solution right off the bat when we solve the ODEs, it would be great.
 

Download Attempt_-_2.mw

Thanks @ecterrab . I tried your suggestion (attached) but it still doesn't work the way I want it. Now c is independent of beta__c, and p__c depends on it, but such that for beta__c=0 I get p__c=0 which is not good. I couldn't really find anything helpful in this case in dsolve,system either.

Download Attempt.mw

@acer Thanks for your code. Although now beta__c appears in both c and p__c, they vanish for beta__c-->0 which they shouldn't. If you check Mathematica's solutions, for beta__c-->0 one gets p__c=A e^(2T) and for c=B/e^(2T) wih A, B being constants, and these clearly are not zero. My guess is that some factors of beta__c are absorbed in c1 and c2 in Maple's solution, but I think this is an unnecessary and unhelpful way of representing the solutions by Maple, at least for my purpose.

Do you think there is a way, for example telling Maple to use a different method of solving ODEs, etc., to make Maple give us the solutions in a different format?
 

@nm Thanks. Yes, I know that Maple's solution is correct which can be confirmed by direct substition. The issue is that one expects that beta__c appears in p__c, and furthermore the solutions do not vanish for beta__c=0, which seemingly is the case in this form that Maple presents them. My guess is that some factors of beta__c are absorbed in c1 and c2 in Maple's solution, but I think this is an unnecessary and unhelpful way of representing the solutions by Maple, at least for my purpose.