1550078481-Ordinary_Differential_Equations__Roberts_

N-th Order Linear Differential Equations

L

I

I

T e

I

-+

iA

Vertical 1

' ' /

' / /

Fg=mg

Figure 4.4 A Simple P endulum

201

The net force that tends to restore the system to equilibrium is Fr. If we let
s represent the arc length AB, then applying Newton's second law of motion
(F =ma), we get

(1)

d^2 s

Fr= mgsinB = - m dt 2

because -d^2 s/dt^2 is the acceleration along the arc AB. Since s =LB,

(2)

Substituting (2) into (1), we obtain

or

(3)

d^2 B

gsinB = -L-dt2

d^2 B

L - 2 + gsinB = 0.

dt

This differential equation is nonlinear due to the function sin B. The expli cit
solution of the DE (3) involves elliptic integrals.