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238                                         EXCEL NUMERICAL METHODS


               Higher-Order  Differential Equations

                   Differential equations of higher order can also be  solved using the methods
               described in this chapter, since a differential equation of order n can be converted
               into  a  set  of  n  first-order  differential  equations.  For  example,  consider  the
               following second-order differential equation (equation 10-30) that describes the
               damped vibration of a mass m connected to a rigid support by a linear spring with
               coefficient k, and a vibration damper with coefficient kd, illustrated in Figure 10-
               15.














                                  Figure 10-15.  A damped vibration system.


                                           d2x      dx
                                        m---+kd-+ksx=O                           (10-30)
                                           dt       dt
                   Equation 10-30 can be rearranged to


                                                                                (1 0-30a)

                   The values of the mass, spring coefficient and damper coefficient are shown
               in Figure 10-16.  We want to calculate the position x of the mass at time intervals
               from t = 0, when the mass has been given an initial displacement of 10 cm from
               its rest position.








                     Figure 10-16.  Parameters used in the damped vibration calculation in Figure 10-17.
                    (folder 'Chapter 10 Examples', workbook 'ODE Examples', worksheet '2nd Order ODE')
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