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APPLICATIONS OF SECOND-ORDER DIFFERENTIAL …

APPLICATIONS OF SECOND-ORDER DIFFERENTIAL EQUATIONSS econd- order linear DIFFERENTIAL equations have a variety of APPLICATIONS in science andengineering. In this section we explore two of them: the vibration of springs and SPRINGSWe consider the motion of an object with mass at the end of a spring that is either ver-tical (as in Figure 1) or horizontal on a level surface (as in Figure 2).In Section we discussed Hooke s Law, which says that if the spring is stretched (orcompressed) units from its natural length, then it exerts a force that is proportional to :where is a positive constant (called the spring constant).

Second Law gives or Equation 3 is a second-order linear differential equation and its auxiliary equation is. The roots are We need to discuss three cases. CASE I (overdamping) In this case and are distinct real roots and Since , , and are all positive, we have , so the roots and given by Equations 4 must both be negative. This shows that as .

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