Please, how can I plot the phase plane and potential function? I mean, what is the code, and which program should be used? Question 17.
752 Chapter 12 Stability of Autonomous Systems In Problems 7-12, use the potential plane to help sketch the phase plane diagrams for the given equations. d-x x-p 0=x6+ dt2 dt2 dx 9. -2x2+x-1=0 dt2 d2x d-x x 10 sinx=0 11. -p d-x 12 -x-13=0 dt2 fixed, sketch the potential function and the phase plane diagram for >0 and also for <0. Describe how the behavior of solutions to the equation differs in these cases. 18. Cusps. We have observed that where the potential energy function G(x) for a conservative system has a strict maximum (minimum), the corresponding critical point is a saddle point center. At a critical point xo,0 for which Gx also has a point of inflection e.g Gxo=gxo=0, Gxo=gxo=0, and Gxo= gxo0, the curve in the phase plane has a cusp. Demonstrate this by sketching the potential plane and the phase plane diagrams for the equation In Problems 13-16, use the energy function to assist in sketching the phase plane diagrams for the given non-conservative systems. 13. dp ip dt2 14 dt2 dx 15 ztp d-x 16. dt2 -0 19. General Relativity. In studying the relativistic motion of a particle moving in the gravitational field of a larger body, one encounters the equation du +u-Xu-1=0 dg2 where u is inversely proportional to the distance of the particle from the body, is an angle in the plane of motion and is a parameter with 0< <1. a Sketch the phase plane diagram for >>- b Sketch the phase plane diagram for 1/4<x<1. c What observations can be made about the motion of the particle in these two cases? ip dx ip x-2 dx +2x+x-1=0 ip 17. Nonlinear Spring. The general nonlinear spring equation dt2 where >0 and are parameters, is used to model a variety of physical phenomena. Holding >0
