1550078481-Ordinary_Differential_Equations__Roberts_

392 Ordinary Differential Equations

Exercise 7. a. Write system (10) as an equivalent system of first-order dif-

ferential equations in matrix notation.

b. Use EIGEN or your computer software to find the general solution of

the resulting system for m 1 = 30 g, m2 = 20 g, f1 = 50 cm, £2 = 25 cm,

and g = 980 cm/s^2.

9.3 The Path of an Electron

In chapter 6, we stated that the position (x, y) of an electron which was
initially at rest at the origin and subject to a magnetic field of intensity H
and an electric field of intensity E satisfied the second-order system initial
value problem

x" = -HRy' +ER

(11a)

y" = HRx'

(11b) x(O) = 0, x'(O) = 0, y(O) = 0, y'(O) = 0

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

Here the independent variable is time, t , and the dependent variables are x
and y. In chapter 7 we showed how to write a system of first-order differential

equations which is equivalent to system (9) by letting u 1 = x, u 2 = x', u 3 = y,

and u4 = y'. Hence, in chapter 7 we found that the following system of four

first-order differential equations is equivalent to (11a)

I

U1 = U2

u~ = -HRu 4 +ER

I

U3 = U4

u~ = HRu2.

Using matrix-vector notation, this system may be rewritten as

(12a)

(

0 1

/ _ 0 0

u - 0 0

0 HR

~ 0 -~R) 1 u+ (iR) 0 ·

0 0 0

The initial conditions are u 1 (0) = 0, u 2 (0) = 0, u 3 (0) = 0, and u 4 (0) = 0. Or

in vector notation the initial conditions are

(12b) u(O) = 0.