1550078481-Ordinary_Differential_Equations__Roberts_

378 Ordinary Differential Equations

and

Y2 ( x) = e^2 x sin x ( -~) + e^2 x cos x ( -~)

_ (-e^2 x sin x -e^2 x cos x)

• e^2 x cosx ·

Furthermore, the real general solution ofy' = Ay is y(x) = kiy 1 (x)+k2y2(x)

where k 1 and k2 are arbitrary real constants.

EXAMPLE 4 Finding the Real General Solution of a

Homogeneous System

Find the real general solution of the homogeneous linear system

(21)

SOLUTION

y ' = ( ~

-1

4

1

1

We ran the computer program EIGEN, setting the size of the matrix to
three and entering the values for the elements of the matrix which appears in
equation (21). The output of t he program is shown in Figure 8.7.

THE MATRIX liJHOSE EIGENVALUES AND EIGENVECTORS ARE TO BE CALCULATED IS

l.OOOOE+oo 4.0000EtOO 3.0000lltOO

O.OOOOEtOO l.OOOOEtOO -1. OOOOEtOO

-1. OOOOEtOO l.OOOOEtOO 2.0000EtOO

AN EIGENVALUE I S l. OOOOOEtOO + 2.00000EtOO I

THE ASSOCIATED EIGENVECTOR I S

- 8.71550E-Ol + l. 232 54Et00 I

l.15057E-0 2 + -4.18516E-Ol I

• 8.3 7032 E-Ol + - 2.30115E-02 I

AN EIGENVALUE IS l.OOOOOEtOO + - 2.00000EtOO I

THE ASSOCIATED EIGENVECTOR I S

-8. 71550E-Ol + -l.23254Et 00 I

l .15057E- 02 + 4. 18516E-Ol I

-8.37032E-Ol + 2. 30115 E-0 (^2) I
AN EIGENVALUE IS 2.00000EtOO + O.OOOOOEtOO I
THE ASSOCIATED EIGENVECTOR IS
5.38516E-Ol + O.OOOOOEtOO I
5.38516E-Ol + O.OOOOOEtOO I
-5.3 851 6E-Ol + O.OOOOOEtOO I
Figure 8.7 Eigenvalues and Eigenvectors of the Matrix in Equation (21)