Advanced Calculus fi..

Chapter 10 Partial Differential Equations 661The discussion to follow applies the theory of linear differential equations asdeveloped in Sections 9.4 to 9.7.10.2 REVIEW OF EQUATION FOR FORCED VIBRATIONS OF A SPRINGWe recall briefly some basic facts concerning Eq. (10.4). Throughout we assumem=O,h=O,k >0.a) Simple harmonic motion. Here h = 0, F(t) = 0, and m > 0. The equationbecomesThe solutions are sinusoidal oscillations:b) Damped vibrations. Here m > 0, F(t) = 0, and h > 0, but h,is small: h2 0, F(t) E 0. The equation readsThe solutions are decaying exponential functions:k '~=ce-~', a=-h '> -(10.11)A similar result is obtained if we consider an equation (10.8) in which h islarge in comparison to m: h2 > 4mk2; in fact, (10.10) can be considered asthe limiting case: m -+ 0 of (10.8).d) Equilibrium. We assume F(t) = Fo, a constant. The equation becomesThe equilibrium vaiue of x is that for which x remains constant; hencedxldt = 0, d2x/dt2 = 0. Accordingly, at equilibrium,