1550078481-Ordinary_Differential_Equations__Roberts_

300 Ordinary Differential Equations

(33a)

(33b)

where g is the constant of gravitational acceleration.

I

~

I I Y2

I. I

1 vertical 1

Figure 6.9 A Double Pendulum

Exercise 3. For m 1 = .3 kg, m2 = .2 kg, £ 1 = .5 m, P2 = .25 m, and

g = 9.8 m/s^2 solve the system of differential equations (33) subject to the ini-

tial conditions y 1 (0) = .0 5 rad, y~(O) = .15 rad/s, y2(0) = .1 rad, and y~(O) =

-.2 rad/s. (HINT: Multiply equation (33a) by £ 2 and multiply equation (33b)

by £ 1. Subtract one of these new equations from the other to eliminate the

term containing y~ as a factor. Solve the resulting equat ion for Y2· Differ-

entiate twice to get y~ and y~. Substitute the expressions for y 2 and y~ into

equation (33b) and obtain a fourth order differential equation in y 1. Use

POLYRTS or your computer software to find the roots of the associated aux-

iliary equation. ·write the general solution y 1 , then find y2, and finally satisfy

the initial conditions.)

The Path of an Electron In 1897 , J. J. Thomso n demonstrated the

existence of the electron by determining the ratio of the charge of an electron

to its mass. Let the ratio be R = q/ m where q is the charge of an electron

and m is its mass. The position (x, y) of an electron in the plane satisfies the

system of differential equations