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differential equation applications 195F (x) kF (x,t) extFigure 9.1 A mass m attached to a spring with restoring force F k (x) and with an externalagency (a hand) subjecting the mass to a time-dependent driving force as well.where F k (x) is the restoring force exerted by the spring and F ext (x, t) is the externalforce. Equation (9.1) is the differential equation we must solve for arbitrary forces.Because we are not told just how the spring departs from being linear, we are freeto try out some different models. As our first model, we try a potential that is aharmonic oscillator for small displacements x and also contains a perturbation thatintroduces a nonlinear term to the force for large x values:V (x) ≃ 1 2 kx2 (1 − 2 3 αx ), (9.2)⇒dV (x)F k (x)=−dx= −kx(1 − xαx)=md2 dt 2 , (9.3)where we have omitted the time-dependent external force. Equation (9.3) isthe second-order ODE we need to solve. If αx ≪ 1, we should have essentiallyharmonic motion.We can understand the basic physics of this model by looking at the curves onthe left in Figure 9.2. As long as x1/α, the force will become repulsive andthe mass will “roll” down the potential hill.As a second model of a nonlinear oscillator, we assume that the spring’s potentialfunction is proportional to some arbitrary even power p of the displacement x fromequilibrium:V (x)= 1 p kxp , ( p even). (9.4)−101COPYRIGHT 2008, PRINCET O N UNIVE R S I T Y P R E S SEVALUATION COPY ONLY. NOT FOR USE IN COURSES.ALLpup_06.04 — 2008/2/15 — Page 195