Which ODE systems can be integrated by quadrature? I am interested if it can be proven the non integrability by quadrature of a particular system of ordinary differential equations. And I read that the differential Galois theory could be the answer, but I want to be sure before I start studying it and realize after several weeks that I have wasted my time.
Suppose you have a nonlinear system of first order of ordinary differential equations:
\begin{eqnarray}
\dfrac{d v}{d r}(r)&=&\dfrac{R\sinh(r)}{r\sinh(R)}\left[1-\zeta_1\sqrt{\sigma^2(r)+2(\sigma(r)-\beta(r))^2}\right]-\epsilon\left[1+\zeta_2\sqrt{\sigma^2(r)+2(\sigma(r)-\beta(r))^2}\right]\\&-&2\dfrac{v(r)}{r},\\
\dfrac{d\sigma}{d r}(r)&=&-\dfrac{2\beta(r)}{r},\\
v(r)\dfrac{d\beta}{d r}(r)&=&2\left(\dfrac{1}{e^{\beta(r)}+2}\dfrac{d v}{d r}(r)+\left(\dfrac{2}{e^{\beta(r)}+2}-1\right)\dfrac{v(r)}{r}\right),
\end{eqnarray}
with $r\in(0,R)$.
It is possible with the differential theory of Galois prove that this system can or can't be integrated by quadratures? If it isn't possible, there is another way to prove this?
Thanks in advance.
 A: *

*Of course there are examples of ODE when you cannot express solutions in elementary functions, the simplest one is
$$
y'=e^{-t^2},
$$
but this is probably not what you are looking for.

*There is the so-called Liouville theorem that states that the equation 
$$
y'=y^2-t^2 
$$
cannot be integrated, that is, the solution cannot be represented by a finite combination of elementary functions and integrals thereof. The proof relies on reducing this equation to s second order linear ODE with non-constant coefficients, for which the differential Galois theory exists. (More details with all the proofs can be found in An introduction to differential algebra by Irving Kaplansky).

*I do not think that there is a general theory of "integrability" for arbitrary nonlinear equations. However, there are specific classes of equations,  when system can be called integrable. I remark, that in general it does not mean you can write your solution as a combination elementary functions. An obvious example is Hamiltonian systems.

*For a particular nonlinear ODE sometimes one can figure out that explicit solution can be found. The theory behind it is the theory of Lie groups (Sophus Lie introduced his groups to solve differential equations). A good reference to look in is Olver's Lie groups and differential equations.

*For a specific system your best guess is to type it into one of the computer algebra system (such as Maple or Mathematica) and ask for an explicit solution. If it does not work, you can try to learn available computer procedures for checking symmetries and use it. But this will already require some time investment.
