1550078481-Ordinary_Differential_Equations__Roberts_

276 Ordinary Differential Equations

value problem

(1) my"+cy'+ksiny=O; y(O)=co, y' (O)=c1

where c ~ O is a constant of proportionality, k = mg/ e, g is the constant of

acceleration due to gravity, c 0 is the initial displacement of t he pendulum,
and c 1 is the initial velocity of the pendulum.

I

I

I

I

Vertical 1

I

bob of

massm

Figure 6.1 Simple Pendulum
The differential equation appearing in (1) is nonlinear because of the factor
sin y. An approximation which is made in order to linearize the differential
equation is to replace sin y by y. This approximation is valid for small angles,
say IYI < .1 radia ns ~ 5.73°. So for y small the following linear initial value
problem approximately describes the motion of a simple pendulum

(2) my"+ cy' + ky = O; y(O) =co, y'(O) = c1.

If there is some external force f(t) acting on t he simple pendulum, such as
a weight or main spring in a clock which is mechanically connected to and
driving the rod of the p endulum, then the linearized homogeneous initial value
problem (2) must be replaced by the nonhomogeneous initial value problem

(3) my" + cy' + ky = f(t); y(O) = co, y' (0) = c1.

Often f(t) is periodic and representable as f(t) = Esinwt where E and ware

constants.

A Mass on a Spring A spring of natural length Lis suspended by one

end from a fixed support. A body of mass m, where m is small compared
to the mass of the spring in order to avoid exceeding the elastic limit of the
spring, is attached to the other end of the spring and the resulting system is
allowed to come to rest at its equilibrium position. Suppose in the equilibrium

position the length of the elongated spring is L + e where e > 0 and e is sm all

compared to L. See Figure 6.2. According to Hooke's law, the elongation

(R > 0) and compression (R < 0) of a spring is directly proportiona l to t he force