EDIT2: After some discussion here's the original problem: Let M be a n-D manifold and $\dot x=F(x)u_1, F\in \mathbb{R}^{n\times m}, u_1 \in \mathbb{R}^{m}$ be a control system evolving on M (F is the system matrix i.e. state transition function, and $u_1$ is the input of the system. For all practical purposes $u_1$ is an m-vector from an input space $\mathbb{R}^{m}$). Now let $x=\Psi (y)$ be a coordinate change on M and $u_2=M(y)u_1$ a transformation of the input $u_1$ of the first system. By applying these maps on the system, you get the new equations $\dot y=F(y)u_2$. As you may notice, F is the same in both systems. The problem is why is this happening i.e. for what systems and transformations does this property hold?
EDIT1: A more interesting story is when the d.e. is a matrix equation. For example: $F(y)=DyF(x)G(x)$ where, $F\in \mathbb{R}^{m\times n}, Dy \in \mathbb{R}^{m\times m}$ (the Jacobian matrix of $y=y(x)$) and $G \in \mathbb{R}^{n\times n}$? Apparently $x,y$ are m-vectors.
i have the following d.e. f(y)={y}'f(x)g(x). Does anybody know the solution or a way to solve this d.e. (maybe it is a know form)? Note that $f,g,x,y$ are all real. Thanks in advance!