Consider a differential equation of the form $$f^{(n)}(z)=P(z,f(z),f'(z),...,f^{(n-1)}(z))$$ where $P$ is a polynomial in $n+1$ variables, with the initial condition (for definitness) $$f^{(k)}(0)=0,\qquad 1\leq k\leq n-1.$$ By the ODE existence theorem, there exists a unique solution $f(z)$ which is holomorphic near the origin of the complex plane. My question is whether it is possible to decide whether $f(z)$ can be analytically continued to the whole complex plane. For example, suppose that the coefficients of P are rational numbers. Is there a finite decision procedure that will take P and determine whether $f(z)$ is entire?