1550078481-Ordinary_Differential_Equations__Roberts_

Systems of First-Order Differential Equations 329

SOLUTION

Let u1 = x, u2 = x', u3 = y, and U4 = y'. So u 1 represents the x-coordinate

of the position of the electron and u 2 represents the velocity of the electron in

the x-direction. While u3 represents the y-coordinate of the electron and u 4

represents its velocity in the y-direction. Differentiating u 1 , u 2 , u 3 , and u 4 ,

substituting from (23a), and then substituting for x, x', y, and y' in terms of

u 1 , u2, u3, and u 4 , we obtain the following first-order system of four equations

which is equivalent to the system (23a).

u~ = x' = U2

(24a)

u; = x" = -HRy' +ER -HRu 4 +ER

u; = y' = U4

I

U4 = y" = HRx' = HRu2.

Substituting for x, x', y, and y' in terms of u 1 , u 2 , u 3 , and u 4 , we see that
the initial conditions which are equivalent to (23b) are

(24b) u 1 (0) = 0, u2(0) = 0, u3(0) = 0, u4(0) = 0.

Hence, the required equivalent first-order system initial value problem consists
of the system (24a) together with the initial conditions (24b).

EXERCISES

l. Verify that {y 1 (x) = 3ex, y 2 (x) = ex} is a solution on the interval

( -oo, oo) of the system of differential equations

y~ 2y1 - 3y2

y; Y1 - 2y2

2. Verify that {y 1 (x) = e-x, y 2 (x) = e-x} is also a solution on the interval

( - oo, oo) of the system in exercise 1.

3. Verify that {y 1 (x) = -e^2 x( cos x +sin x), Y2(x) = e^2 x cos x} is a solution

on ( -oo, oo) of the system of differential equations

Y~ Y1 - 2y2

y; Y1 + 3y2

4. Verify that {y 1 (x) = 3x - 2, y2(x) = -2x + 3} is a solution on (-oo, oo)

of the system initial value problem

Y~ Y1 + 2y2 + x - 1

y; 3y1 + 2y2 - 5x - 2

Y1 (0) = -2, Y2(0) = 3