Projective sets and ordinary differential equations

We prove that for $n \geq 2$ the set of Cauchy problems of dimension $n$which have a global solution is $\boldsymbol\Sigma_{1}^{1}$-complete and that the set of ordinary differential equations which have a global solution for every initial condition is $\boldsymbol\Pi_{1}^{1}$-complete. The first result still holds if we restrict ourselves to second order equations (in dimension one). We also prove that for $n \geq 2$ the set of Cauchy problems of dimension $n$which have a global solution even if we perturb a bit the initial condition is $\boldsymbol\Pi_{2}^{1}$-complete.