Is the flow of a vector field independent of coordinates?

Question

Assume we have a vector field $X=a_i(x)\frac{\partial}{\partial x_i}$ in local coordinates of a manifold M. Then we can solve an ODE to get the flow of the vector field. If we have in some other coordinates $y_i$, $X=b_i(y)\frac{\partial}{\partial y_i}$, will we end up with the same flow by solving an ODE as in the previous coordinates?