In each of Problems 12 through 16, a second-order equation of the form x'' + f (x, x') = 0, corresponding to a certain mass-and-spring system, is given. Find and classify the critical points of the equivalent first-order system.

x'' - 8x + 2x³ = 0: Here the linear part of the force is repulsive rather than attractive (as for an ordinary spring). Verify that the critical points resemble those shown in Fig. 9.4.12. Thus there are two stable equilibrium points and three types of periodic oscillations.

Transcribed Image Text:

-2 IN 2 X FIGURE 9.4.12. The phase portrait for Problem 14.