Consider the \(\operatorname{ODE} \partial_{t} \xi(t, x)=x+b(\xi(t, x)), \xi(0, x)=x\), where \(b \in \mathcal{C}_{b}^{2}\left(\mathbb{R}^{d}, \mathbb{R}^{d}ight)\) such that \(\left|\partial_{x}^{\alpha} b(x)ight|,|\alpha|=2\), is Lipschitz continuous. Show that all \(\partial_{x}^{\alpha} \xi(t, x),|\alpha| \leqslant 2\), are polynomially bounded.

Assume that the derivatives \(\partial_{x}^{\alpha} \xi(t, x)\) exist and solve the formally differentiated ODE.