TITLE: Decidability of differential equations
QUESTION [6 upvotes]: Is there anything well-known about the algorithmic decidability of the satisfiability of an ODE $\dot{x}=f(x)$, $x: [0,1]\to R^n$ with an initial condition $x(0)=x_0$, given that $f(x)$ belongs to some specified class of functions -- for example, component-wise polynomial?

REPLY [3 votes]: In the the paper Boundedness of the Domain of Definition is Undecidable for Polynomial ODEs by Daniel S. Graça et al. (preprint version) they prove the undecidability of deciding the boundedness of the maximal domain of solution of a polynomial ODE.