Here are some minor steps, mostly to simplify the presentation of the problem. I don't think that they'll have a major impact towards obtaining a symbolic solution (but every little bit helps): 

1. Please get rid of the subscript t. It's just clutter that makes the problem harder to read.
2. For the same reason, replace 1-gamma with some other variable, say g.
3. Substitute 1 - phi__1 - phi__2 for phi__3 in the objective.
4. Step 3 eliminates all decision variables from the fourth term, so that term can be removed entirely; it can't change the maximizing values of the decision variables.
5. Replace a__1 + 1 by A__1, a__2 + 1 by A__2, b__1 - 1 by B__1, b__2 - 1 by B__2, c__1 + 1 by C__1, c__2 + 1 by C__2, d__1 - 1 by D__1, d__2 - 1 by D__2, and w+1 by W.

With these changes, the objective can be entered like this:

phi:= <phi1,phi2>: A:= <A1|A2>: B:= <B1|B2>: C:= <C1|C2>: E:= <D1|D2>:

One way:
Ob:= (pi__w*((W-<2|2>.phi)^g+beta*(pi__a*(A.phi-1)^g+(1-pi__a)*(C.phi-1)^g))
    +(1-pi__w)*beta*(pi__b*(B.phi+1)^g+(1-pi__b)*(E.phi+1)^g))/g;

Or, using more-Maple-specific operators:
Ob:= (<map(`.`, [-<2|2>, A, C, B, E], phi) + [W,-1,-1,1,1]>^%T)^~g
    . <pi__w*~(1,beta*~(pi__a,1-pi__a)),(1-pi__w)*beta*~(pi__b,1-pi__b)>/g; 

Now here's what may allow some major progress towards a symbolic solution: I suspect that there are some bounds for your variables and constants that you haven't stated. Which are necessarily nonnegative? Which are necessarily in the interval 0..1?

@rcorless In addition to the errors that you've correctly pointed out, the code also has both k and k[1] used as bound^[*1] variables of summation. So k is being used four ways.

[*1] Bound here is the adjective, the opposite of free and the past participle of to bind. It's not meant in its noun sense related to the boundaries or limits of summation. 

@tomleslie Good Answer; voted up. It can be made a bit simpler because Unit is pre-defined as a top-level command. It does pass directly to Units:-Unit, but you don't need to explicitly invoke the Units package to use it.

@pik1432 The code that I gave must be done before you use with(DEtools). (That's why I said "Then proceed with the code that you already had"---though I do realize that I should've stated that more clearly; so, sorry about that.)

Since your curve is not a function in the usual mathematical sense (exactly one y-value for every x in its domain), it's not clear what you mean by "area under this function".

@pik1432 

DEtools[diffop2de] was not written to work on equations. This is a simple issue of syntax rather than a mathematical reason why it doesn't work. So, it's very easy to overload it so that it will work:

unprotect(DEtools):
DEtools[diffop2de]:= L-> 
    `if`(L::algebraic, `DEtools/diffop2de`, curry(map, procname))(args):
protect(DEtools):

Then just proceed with the code that you already had.

What is the origin of this problem? It puzzles me that someone took the trouble to come up with the interesting coefficients 1/3, 1/9, 2/3, 4/9 even though they play no role in the solution.

And why do you want to use RK2 specifically? This problem too trivial to highlight any differences between methods. Most numerical ODE solvers (including all the RKs) start by algebraically solving the system for the highest-order derivatives. In this case, that gives

diff(A[0](t), t) = 2*A[2](t),
diff(A[1](t), t) = 0, #for all t
diff(A[2](t), t) = 0 #for all t

@Rouben Rostamian  The three minima that she^[*1] is referring to are the critical points of the restriction of the objective function to a steepest-descent line, not critical points of the objective itself. Using the initial point (x,y) = (-1,1), this restricted objective is P:= t-> 16*(2*t - 1)^2*(1600*t^2 + 1). This has three critical points: two minima and one maximum. So, yes, the statement that there are three minima is indeed incorrect, but I don't think that it's incorrect in the way that you were thinking. There is still a need to pick one of them.

[*1] I guess that the OP is a woman based on the name Zeineb, which is a female name of Arabic origin.

@Rouben Rostamian The objective function here is the Rosenbrock function (see Wikipedia link), which is a standard test case for numerical minimization techniques. It's intentionally designed to have a unique local (and also global, but that's not as relevant) minimum that's trivial to compute symbolically "by hand" yet difficult numerically, especially for gradient-based techniques. Thus it's clear to me that the OP is attempting to code and test a numerical technique and is not a student taking a first course in multivariable calculus (where that "by hand" method is taught).

@Axel Vogt Sorry, I wasn't trying! to correct you. My interpretation was wrong also. I now see that the OP has the confusing habit of putting the expressions to be input on the right sides of equations and the desired output on the left. The OP also consistently mentions the desired output before mentioning the input in every sentence where both are mentioned. Both of those habits are the reverse of the usual writing style. So, I asked "Do you have it reversed?" But it turns out that you had the direction of conversion that the OP intended. It just needs to be converted from regular BesselJ to SphericalBesselJ, which seems pretty easy, although we'd need to define SBesselJ as a new symbolic mathematical function.

@Carl Love I can't think of any good reason to not split that 3rd subsequence in two: Replace f[3] with

f[3]:= n-> 4*n-1:
f[4]:= n-> 4*n:

That should make the analysis easier. The fact that these two appear so close to each other on the graphs is not a good reason to keep them together when it's obvious that 4-cycles are key to understanding the overall sequence.

@anthei If you want to use Logic:-Dual in your version of Maple, you can replace it with a corrected procedure, like this:

restart:
unprotect(Logic:-Dual):
Logic:-Dual:= proc(e)
uses L= Logic;    
local
    o,
    O:= [
        `&and`, `&or`, `&not`, `&iff`, 
        `&xor`, `&nand`, `&nor`, `&implies`
    ],
    D:= [
        true= false, L:-`&and`= L:-`&or`, 
        L:-`&iff`= L:-`&xor`, L:-`&nand`= L:-`&nor`
    ]
;    
    evalindets(
        eval(
            subs(seq(cat(``,o)= L[o], o= O), e),
            [
                D[], (rhs= lhs)~(D)[], 
                L:-`&implies`= ((p,q)-> L:-`&not`(L:-`&implies`(q,p)))
            ]
        ),
        (L:-`&not`@@2)(anything), op@@2
    )
end proc:
protect(Logic:-Dual):

@Axel Vogt My interpretation of the Question is that the desired direction of conversion is from BesselJ to an elementary function.

What Maple version are you using? In Maple 2019, this conversion happens automatically (here using n=9/2 for example):

BesselJ(9/2, x);
             

@anthei Maple 18 is not the same as Maple 2018. The chronological sequence of annual Maple version numbers since Maple 10 has been 10, 11, ..., 17, 18, 2015, 2016, ..., 2021, 2022.