# Differential equation with different initial condition

Given an vector field $$v:U (\subset R^n)\to{R^n}$$. Consider the differential equation $$\dot{x} = v(x)$$. It's given that this differential equation with an intial condition $$x(0) = x_0$$ has unique solution for all $$x_0$$ in U.

Let $$\Phi_{x_0}$$ be the solution with initial value $$x_0$$. Is it true that $$\Phi_{x_0+h} = \Phi_{x_0}+h$$ ? If not, is there a way to prove that the function $$\Psi(x,t) = \Phi_{x}(t)$$ is smooth, given that $$v$$ is smooth vector field and U has no singular points of $$v$$, i.e $$v(x) \neq 0$$ for all x in U, using rectification theorem ?

• The answer to the first question is no and a point where the vector field vanishes is called a critical point, not singular point. See this article: en.wikipedia.org/wiki/… – jobe Feb 27 at 15:49