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9Differential Equation ApplicationsPart of the attraction of computational problem solving is that it is easy to solve almostevery differential equation. Consequently, while most traditional (read “analytic”)treatments of oscillations are limited to the small displacements about equilibriumwhere the restoring forces are linear, we eliminate those restrictions here and revealsome interesting nonlinear physics. In Unit I we look at oscillators that may beharmonic for certain parameter values but then become anharmonic. We start withsimple systems that have analytic solutions, use them to test various differentialequationsolvers, and then include time-dependent forces and investigate nonlinearresonances and beating. 1 In Unit II we examine how a differential-equation solvermay be combined with a search algorithm to solve the eigenvalue problem. In Unit IIIwe investigate how to solve the simultaneous ordinary differential equations (ODEs)that arise in scattering, projectile motion, and planetary orbits.9.1 Unit I. Free Nonlinear OscillationsProblem: In Figure 9.1 we show a mass m attached to a spring that exerts a restoringforce toward the origin, as well as a hand that exerts a time-dependent externalforce on the mass. We are told that the restoring force exerted by the spring isnonharmonic, that is, not simply proportional to displacement from equilibrium,but we are not given details as to how this is nonharmonic. Your problem is tosolve for the motion of the mass as a function of time. You may assume the motionis constrained to one dimension.9.2 Nonlinear Oscillators (Models)This is a problem in classical mechanics for which Newton’s second law providesus with the equation of motionF k (x)+F ext (x, t)=m d2 xdt 2 , (9.1)1 In Chapter 12, “Discrete & Continuous Nonlinear Dynamics,” we make a related studyof the realistic pendulum and its chaotic behavior. Some special properties of nonlinearequations are discussed in Chapter 19, “Solitons & Computational Fluid Dynamics.”−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 194