...is generally not calculable in "closed" form. It can only be solved numerically or with the help of a famous series (Riemann Zeta). The limits of integration must be taken into account. For example, when integrating by parts, one ends up with ln(sin(y)), and the "ln" exists in the real world only for sin(y)>0. Gradstein/Ryshik, Volume 1, page 465, provides two typical examples.

...the ode has no solution for the specific initial value. This is probably not due to Maple, but rather to the theory of odes.

@nm 

...substituting u = y^2 and solving for u´, the differential equation becomes u´ = 1/(t^3+t)-u/t. This is the explicit form used in existence and uniqueness proofs for solutions of such equations. In u´ = f(t,u), f(t,u) is continuous in a domain {(t;u)}, and the Lipschitz condition holds after restricting the domain near the origin. Then a solution exists (Peano, Picard/Lindelöf, ...). However, the initial value (0;0) does not lie in this domain. Therefore, the differential equation has no solution for this specific initial value near the boundary of the domain.

BTW: And continuous approximations in the domain of continuity toward the initial value (0;0) should be treated with caution and usually yield errors. The topic of "convergence on the boundary" is a very difficult one, especially for series in the complex plane.

The ODE in question is of the "Jacobi" type. There are many solutions available on the Internet. Last but not least, "Kamke" on page 335, I250, provides sophisticated instructions for developing a closed solution.

The "Legendre proof" by vv is great and once again instructive for me as a Maple beginner. A completely different solution, which I unfortunately cannot yet formulate in Maple, uses Fermat's principle of infinite descent. Here is just the idea of ​​the proof: (I occasionally have to put my associated paperwork into a reasonable form)

There is an error in the following, so forget it...

The term to be examined can be transformed into a product of a binomial coefficients. Their denominators (in written form as n!/(k!*(n-k)!) ) form a decreasing sequence of natural numbers. The last factor contains either 0! or 1! in the denominator. The term to be examined is therefore a natural number as a product of binomial coefficients.

Since I only recently got to know Maple, here is just a suggestion that uses numerical force. I can't program this in Maple yet:
1.) Choose a natural number n and thus a set of numbers 1, 2, ..., n.
2.) Use the binomial coefficient to determine all subsets of six numbers.
3.) Use Euler's volume formula (or the equivalent determinant form) to calculate the volume V.
4.) Remove all subsets with V less than or equal to zero.
5.) Remove all other subsets that do not contain a prime number. Repeat this process for the rest.
6.) Determine the subset of six numbers for which V is minimal. In my opinion, that would be a solution.
7.) If no solution has been found for n, start a new search from 1. with set n+1, ...2*n, etc.

A homogeneous ODE is present. The term "ode" is factorable in y(x). To exclude the zero solution, it should be factored out and only the remaining factor should be treated as a homogeneous ODE.

If I understand you correctly, you have to solve a linear homogeneous system of equations. For a solution that is not zero, your determinant must therefore be zero. If the interval in which beta would lie is limited, then only numerical force will help (personal experience). To do this, I would break the interval down into support points and specify beta step by step and then calculate the determinant in each case. The position of the zero point can be estimated from the table for (beta, det). The process is then repeated with finer subdivisions as nesting of the beta value.

Put the exponential function exp(x) as a factor in front of the integral, then differentiate both sides of the equation, calculate y´(0), solve the unwieldy ordinary differential equation of second order

Your equation is again a Painleve equation (https://en.wikipedia.org/wiki/Painlev%C3%A9_transcendents). It would be classified as type 2 here, but with unpleasant coefficients. These equations can only be solved in closed form in exceptional cases (e.g. homemade examples). A generally valid solution method is not known. Therefore, experience shows that approximation methods are used. This can be done numerically after transformation into a first-order system or using an approach in the sense of Ritz/Galerkin and minimizing the equation defect.
Please see the literature references provided in the link.

It is an equation named after Painleve with the structure
y´´+a*y^3+b=0. For more information see "Kamke" (especially the book from 1948), or the German-book, page 544, no. 6.9 and 6.10.

Depending on the coefficients, only special cases can be solved in a good-natured way.

From my own experience, it would be worth trying to use the Ritz/Galerkin method. This makes finding a solution to the equation a minimal task, which Maple seems to be good at. There is a lot of literature on the Ritz/Galerkin method. I will just say this: A plausible approach to a solution function with free parameters to be calculated is chosen, e.g. as a linear combination. The so-called equation defect must then be minimized. This is basically a positive definite measure for the difference "left side minus right side" of the equation.