1550078481-Ordinary_Differential_Equations__Roberts_

Applications of Systems of Equations 481

where c ;:::: 0 is a constant of proportionality, k = mg/£, g is the constant

of acceleration due to gravity, co is the initial angula r displacement of the

pendulum from the downward vertical, and CI is the initial angular velocity

of the pendulum.

Linearized Simple Pendulum with No Damping and No Forcing

Function

Ifwe assume there is no damping (c = 0) and no forcing function (J(t) = 0) ,

then the differential equation of (1) becomes

my II +Tsmy= mg •^0.

Dividing this equation by m and choosing £ = g in magnitude, we obtain the
nonlinear differential equation

(2) y" + sin y = 0.

This equation is nonlinear because of the factor sin y. The Taylor series ex-
pansion of sin y about y = 0 is

. y3 y5 y7
sm y = y - - + - - - + · · ·.
3! 5! 7!

From this expansion, we see that an approximation which can be made in
order to linearize the differential equation (2) is to replace sin y by y. This
approximation is valid only for small angles, say IYI < .1 radians= 5.73°. So
for small y, the solut ion of the linear initial value problem

(3) y" + y = O; y(O) = co, y' (0) = CI

approximately describes the motion of a simple pendulum. Letting YI = y
and y 2 = y', we can rewrite the initial value problem (3) as the following
equivalent vector initial value problem

(4)

In system (4), YI is the angular displacement of the pendulum and y 2 is its
angular velocity. Thus, YI(t) is the solution of the IVP (3) and y2(t) is the
derivative of the solution. Simultaneously setting Y2 = 0 and -yI = 0 , we find
that the only critical point of system (4) is the origin. The general solution
of (4) is
YI ( t) = Co cost + CI sin t

Y2 (t) = -co sin t + cI cost.