But nobody knows what you have tried. Just saying that you have tried lots of things helps nobody to help you, because that forces someone to write the solution completely for you. In turn, that MIGHT take a great deal of time, because they will need to spend a fair amount of time understanding what you have, what you are trying to solve, what the variables mean, etc.
It appears that you are looking for a solution of this, wanting to find functions g_a(theta) & g_b(theta), for any value of r, as well as the other constants. Clearly there is no solution at r==0, a singularity. And we are given no information about how this process evolves if r were to vary, so effectively we can treat r as a constant, a non-zero one at that since these equations will have no solution at all at the singularity. (You might eventually look at a limiting behavior as r approaches zero though. This is also beyond the scope of Answers, as not being a MATLAB question.)
So one thing you can do is multiply (53) by r^(-1/2). That will simplify it by a fair amount. Or you might try multiplying by just r^(1/2). Again, that will simplify things a bit.
Similarly, do the same for (54). I can't know which is more appropriate, but either might work as well as the other.
I would also be looking for a non-dimensionalization of the problem.
Thus, I have NO idea what h, l, u all mean in context here. What are the units of h, l and u? I might guess that all three have similar units. In fact, I might even guess that these variables have units of length, since they appear in ways where they might also cancel the units of length on r. And since we are told that r is a coordinate variable, it has units of length.
For example, I see the term
Since the constant 1 (one, not l) has no units at all, it makes no physical sense to add h/u to 1, UNLESS h and u have the same units, and therefore h/u is also non-dimensional, thus unit-less itself.
So I would suggest it MIGHT make sense to transform the problem in terms of some fixed ratio for h/u. I might also look at non-dimensionalizing r, so work in terms if r/l or r/h, or even r/u. The correct choice depends up on what all of these variables mean to you, in context.
All of this simplifies the problem. It MIGHT help you to understand some characteristics of the solution, much like understanding how Reynolds number impacts the solution of problems where it applies.
Does any of this apply to how you would solve the problem in MATLAB, or if a solution exists at all? Of course not, beyond possibly making the problem a bit less complex. In fact, nothing about the above comments is at all related to MATLAB. But it will simplify the problem a bit. As well, it will improve your understanding of the problem.
In the end, I might be surprised if an analytical solution exists at all. Or not. That won't happen for me without far more understanding of what you are trying to solve than I am willing to invest the time to gain.
Remember that equations, variables, etc., mean little by themselves. So not until you provide sufficient context and understanding can someone really help you here. And that makes this into something that can only become a long, drawn out consulting issue, but not a question about MATLAB.
Other things you can do are to consider other ways of attacking the problem. For example, can you represent g_a(theta) & g_b(theta) in the form of a series in theta? Think Fourier series, or Chebyshev polynomials for example, or other families of functions that would be appropriate. Again, we are given no context at all, so I am just throwing out some random ideas that MIGHT make sense in the proper context.
The point is, you have much that you can do in the way of mathematical analysis worth investing, BEFORE you just throw this at a differential equation solver and hope a solution magically pops out the end. Again though, that makes this not yet a problem about MATLAB, but a problem of mathematics.