1550078481-Ordinary_Differential_Equations__Roberts_

386 Ordinary Differential Equations

where /3 = 2.66316, "( =. 712450 , and

(

0 .3 75485 )

0.000000

-0.0 347007 )

0.000000

(

0.213163)

0.000000

v 3 = 1.38394 ' and

0.000000

(

0.000000 )

0.999978

0.000000 )

-0.0924135

(

0.000000)

0.151868

V^4 = 0.000000.

0.985988

EXAMPLE 2 Solving an Initial Value Problem for System ( 4)

Find the solution to the initial value problem consisting of the system (4)

and the initial conditions u 1 (0) = 1 cm, u2(0) = 0 cm/s, u 3 (0) = -2 cm, and

u4(0) = 0 cm/s.

SOLUTION

Written in vector notation the given initial conditions are

Evaluating equation (5)- the general solution to system (4)- at x = 0, we

see that the const ants c 1 , c2, c3 and c 4 must satisfy

or equivalently

u(O) = C1 V1 + C2V2 + C3V3 + C4V4 = 0

.3 75485 c1 + .213163c3 = 1

.999 978 c2 + .151868c4 = 0

• .0347007c1 + 1.38394c3 = -2


• .0924135c2 + .985988c4 = 0.

Solving the first and third equations of this set for c 1 and c 3 , we find c 1 =

3.43474 and c3 = -1. 35903. And so lving the second and fourth equations of

this set for c2 and c4, we find c2 = C4 = 0. Hence, the so lution of the system (4)
subject to the initial conditions given in this example is equation (5) with

c1 = 3.43474, c2 = 0, c3 = -1.35903, and c 4 = 0. Thus, the solution is

(

u1(x)) ( 1 .28 069 cos/3x - .289 695 COS"fX)

u(x) = u2(x) = -3 .43466 sin/3x + .206393 sin"(x

u3(x) - .11 9188 cosf3x - 1 .88082 cOS"fX

u4(x) .3 17416 sin/3x + 1 .33999 sin"(x