Differential Equations with Special Functions: Are analytical solutions possible?

Question

I want to ask a question to specialists, I am very interested in getting their advice and opinion. Let's take elliptic integrals and hypergeometric functions and write down any differential equations in which they enter. For example, these:

$x'=-EllipticF(x+1|-1)-EllipticF(\sin (x)|1)$

$x'=_{1}F_{1}(x,2,-x)$

Such ODE's are solved numerically:

s1 = NDSolve[{x'[t] == -EllipticF[Sin[x[t]], 1] + 
      EllipticF[-(x[t] + 1), -1], x[0] == 2}, x, {t, 0, 30}];

Plot[Evaluate[x[t] /. s1], {t, 0, 30}, PlotRange -> All, 
  ImageSize -> Small];

s2 = NDSolve[{x'[t] == Hypergeometric1F1[x[t], 2, -x[t]], x[0] == 1}, 
   x, {t, 0, 30}];

Plot[Evaluate[x[t] /. s2], {t, 0, 30}, PlotRange -> All, 
  ImageSize -> Small];

But something else interested me.

1. Are there any approaches that allow solving such ODE's analytically for an arbitrary structure, or for some class of structures (it is desirable to do all this without expansion in a series, so that the solution does not lose a significant part of the properties)?

2. Are there approaches (criteria, transformations) that make it possible to somehow evaluate the internal structure of such ODE's and draw a conclusion about their solvability/unsolvability?

Please, prompt the literature which will allow to study these questions. I would be happy and grateful for support.