Please solve this problem for Question1,3,4 and 6, Thank you!

Question 1 Consider the following nonlinear differential equations = In(r2 + 1) -1, (1) dt (2) dt = sin(x) (x2 - 5r + 6). (3) For each case, find all fixed points, discuss their stability by using the geometric approach (i.e., the plot of the velocity in the (x, i) plane), and sketch the corresponding flow (vector field) on the real line. In addition, sketch the graph of the solution x(t) = X(t, co) versus t in each case and for different initial conditions To- Question 2 Use linear stability analysis to classify the fixed points of equation (2). Question 3 Set an arbitrary initial condition r(0) = To E R. Does the solution to equation (1) blow up in a finite time? Or it exists and it is unique for any finite t > 0 (global solution)? Justify your answer. Question 4 Provide an approximate plot of forward flow map X(t, ro) generated by equation (1) versus co at different times, including t =0. What happens when t -> co? Question 5 Find a smooth velocity field f(x) so that the phase portrait generated by the equation dx/dt = f(x) is consistent with the following one: -2 3 Question 6 Find a potential V(x) for the vector field defined by equation (3)