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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 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?

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?

Are there known necessary and sufficient conditions that specify in terms of an algorithm in a real arithmetic model when a 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 partial results?

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 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?