1550078481-Ordinary_Differential_Equations__Roberts_

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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.

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