文档介绍：5 Applications of Higher-Order Differential Equations In Chapter 4, we discussed several techniques for solving higher-order differential equations. In this chapter, we illustrate how some of these methods can be used to solve initial-value problems that model physical situations. Harmonic Motion Simple Harmonic Motion Suppose that a mass is attached to an elastic spring that is suspended from a rigid support such as a ceiling. According to Hooke’s law, the spring exerts a restoring force in the upward direction that is proportional to the displacement of the spring. Hooke’s Law: F ? ks, where k> 0is the constant of proportional- ity or spring constant, and sis the displacement of the spring. Aspring has natural length b. Whena mass is attached to the spring, it is stretched sunits past its natural length to the equilibrium position x ? 0. When the system is put into motion, the displacement from x ? 0at time tis given by x ? t ?. By Newton’s Second Law of Motion, F ? ma ? md 2x/dt 2, where mrepresents mass and arepresents acceleration. If we assume that there are no other forces 321 322 Chapter 5 Applications of Higher-Order Differential Equations acting on the mass, then we determine the differential equation that models this situation in the following way: m d 2 x dt 2 ???forces acting on the system ??? k ? s ? x ?? mg ?? ks ? kx ? mg. At equilibrium ks ? mg , so after simpli?cation, we obtain the differential equation m d 2 x dt 2 ?? kx or m d 2 x dt 2 ? kx ? 0 . The two initial conditions that are used with this problem are the initial displace- ment x ? 0 ??Α and initial velocity dx/dt ? 0 ??Β. Hence, the function x ? t ? that describes the displacement of the mass with respect to the equilibrium position is found by solving the initial-value problem ?? m d 2 x dt 2 ? kx ? 0 x ? 0 ??Α, dx dt ? 0 ??Β. () The differential equation in initial-value problem () disregards all retarding forces acting on the motion of the mass. The solution x ? t