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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')