Are there known necessary and sufficient conditions that specify in terms of an algorithm in a real arithmetic model (where real operations, elementary functions, and comparisons are elementary steps) when a first order differential equation $x'(t)=F(x(t),t)$ with a real-valued rational function $F(x,t)$ in real scalars $x$ and $t$ is exactly solvable by elementary functions and finitely many integrations?

Useful (positive and negative) partial results? In particular, are there strong sufficient conditions for rational functions with low numerator and denominator degree?

e.g.belonging to a field where you can decide the equality to $0$ with a halting turing machine) and «exactly solvable» (I assume you mean integrability by quadrature in the sense of Liouville). Even being granted the two defaulted meanings above, I fear that there are no known answer to your question, although it is believed by some that the Poincaré question (solvable by rational first-integral) is undecidable. The key-word is differential Galois theory. $\endgroup$ – Loïc Teyssier Mar 17 '15 at 15:24