Exercise 1 Consider the following 1-form: w= (1-2)dx-2112. 1) Is there any function h such that dh=w? 2) Consider the following vector field: a f=27 +(1-2) Or Compute w(fi), then give a vector field that spans the Kernel of w. 3) Let () Find a function h(1.) such that dhw. What is the relationship between the level curves of h and the integral curves of fi? 4) Let the following vector field: Show that a ---(+) ((1-2)-2015). --Vh where Vh is the gradient of h. 5) Let Vh and compute LV to show that the singularity (-.0) of f₂ is asymptotically stable. What about its second singularity (.0)? 6) Are the singularities of fi stable, asymptotically stable or unstable? Exercise.2 1) Find the state representation of the following differential equation: +6y+ay+Buy cos(wt) 2) Let the following vector field: f(x) a (ax+6x+8) what is the link between it and the previous differential equation? 3) Find the singularity of the following dynamics = f(x). 4) Study the stability of those singularities. 5) Set =0 for i = f(x) and find the matrix A such that f(z) = Az. Give the Laplace transformation of this linear dynamical system. 5. 6) Discuss the solution of this linear dynamical system based on the values of a and