Consider the following nonlinear differential equation

dx/dt=ln(x²+1)1(1)

a. find all fixed points, discuss their stability by using the geometric approach (i.e., the plot of the velocity in the (x,) plane), and sketch the corresponding flow (vector field) on the real line. In addition, sketch the graph of the solutionx(t) =X(t,x0) versustin each case and for different initial conditionsx0.

b.Set an arbitrary initial conditionx(0) =x0R. Does the solution to equation (1) blow up in a finite time? Or it exists and it is unique for any finitet0 (global solution)? Justify your answer.

c.Provide an approximate plot of forward flow mapX(t,x0) generated by equation (1) versusx0at different times, includingt= 0. What happens whent ?