I am taking a course in dynamical systems and the Flow Box Theorem was presented to me. The theorem said:
Theorem (Flow Box): In the neighborhood of a regular point ($f(x_0) \neq 0$), a $C^k$ system of linear differential equations is always $C^k$-conjugated (conjugated with the corresponding homeomorphism in $C^k$) to the system of linear differential equations $y'_1=1, \, y'_2 = 0, \, \dots \, ,y'_n=0$
I am trying to prove it finding the corresponding homeomorphism restricted to the neighborhood of the regular point but I am encountering difficulties. Could anyone help me? Thank you very much!