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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!

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    Jon please notice that $J is n \times n$ and $ M is m \times m$ thus $J \neq M$. Furthermore, is nonlinear, for a fact. There is a specific example of this with$F$being the kinematic equations of a unicycle robot.2011-12-18

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This equation can be integrated in in the following way

$\int\frac{dy}{f(y)}=\int dx\frac{1}{f(x)g(x)}+C$

Once the forms of $f$ and $g$ are known, the integrals could be computed.

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    Hamilton equations are first order equations of course. You have just to prove that your transformation is canonical provided $F$ is describing a Hamiltonian system. Check http://en.wikipedia.org/wiki/Hamiltonian_mechanics and http://en.wikipedia.org/wiki/Canonical_transformation. You should not need such a cumbersome condition. Maybe, it would be helpful if you would state the original problem in your question.2011-12-17