(Solved) : Also Show Work Part Let Us Consider System Dx Dt 2xy Dy Dt Y2 X2 1 5 Points Con Rm Equili Q36921619 . . . .

Can you also show the work

Part I. Let us consider the system

dx/dt = 2xy

dy/dt = y2−x2                                                                  (1)

a)(5 points) Confirm that the only equilibrium of the system isthe origin and that Jacobian at this equilibrium has botheigenvalues zero. Explain why x = 0 is an invariant line of (1) andindicate the behavior of solutions on this line;

b)(15 points) Prove that if y doesn’t = 0 we have:

dy/dx = 1/(2x)*y−x/2 *1/y                                                                          (2)

This is a so-called Bernoulli type equation (details arepresented in Appendix A of our manual). Show that the change ofvariable u = y2 leads to the equation:

Du/dx –1/ x *u = −x                                                                                       (3)

c)(20 points)