# a question on visualizing a state of DAE and a question from a continous time nonlinear dynamics

1. Could someone explain me what is the fundamental difference between the dynamical system of the kind $$\dot x = f(x)$$ and $$E \dot x= f(x)$$ where $$E$$ is an matrix with real entries. For the first system, a state is $$x(t)$$ at time $$t$$; is it the same for the second system? I wish to visualise trajectories of such type of dynamical system to be more precise.

2. Consider a continuous time dynamical system \begin{align} \dot x= f\left(x\right) \end{align} on a state-space $$\mathcal{X}$$, where $$x$$ is a coordinate vector of the state, $$f$$ is a non-linear vector valued smooth function of the same dimension as its argument $$x$$.

Could anyone tell me why assume $$f$$ to be smooth and what it means by ''function of the same dimension as its argument $$x$$'', what is a dimension of function, by the way, and why its dimension needs to be equal as $$x$$?

Thank you for your clarification.