1550078481-Ordinary_Differential_Equations__Roberts_

Applications of Linear Systems with Constant Coefficients 385

SOLUTION

For the given values of m 1 , m2, k 1 , and k 2 system (3) b ecomes

(4) u' = (-~

1 0

0 1

0 0

.6 0 -.6

We used the computer program EI GEN to find the eigenvalues and associated

eigenvectors shown in Figure 9.2.

THE MATRIX WHOSE EIGENVALUES AND EIGENVECTORS ARE TO BE CALCULATED IS

O.OOOOEtOO LOOOOEtOO O.OOOOEtOO O.OOOOEtOO

• 7.0000EtOO O.OOOOEtOO L OOOOE+OO O.OOOOEtOO
  O.OOOOE+OO O.OOOOEtOO O.OOOOE+OO L OOOOEtOO
  6.0000E-01 O.OOOOEtOO -6.0000E-01 0. 0000EtOO

AN EIGENVALUE I S O.OOOOOEtOO + 2.66316EtOO I

THE ASSOCIATED EIGENVECTOR IS

3.75485E-Ol + O.OOOOOEtOO I

O.OOOOOE+OO + 9.99978E-Ol I

-3.47007E-02 + O.OOOOOE+OO I

O.OOOOOEtOO t -9.2 41 3 5E-02 I

AN EIGENVALUE I S 0.00000EtOO + -2.66316Et00 I
THE ASSOCIATED EIGENVECTOR IS
3.75485E- 01 + O.OOOOOE+OO I
O.OOOOOEtOO + -9.99978E-Ol I
-3. 47007E- 02 + 0. 00000EtOO I
0. 00000EtOO + 9.24135E-02 I
AN EIGENVALUE IS O.OOOOOEtOO + 7.1 2450E- 01 I
THE ASSOCIATED EIGENVECTOR IS
2.13163E-01 + O.OOOOOEtOO I
O.OOOOOEtOO + l .51868E-Ol I
L 38394Et00 + O.OOOOOEtOO I
O.OOOOOEtOO + 9. 8 5988E- Ol I
AN EIGENVALUE I S O.OOOOOEtOO t -7. 12450E- Ol I
THE ASSOCIATED EIGENVECTOR IS
2.13163E-Ol + O.OOOOOEtOO I
O.OOOOOEtOO + -l.51 868 E-Ol I
L 38394Et00 + O.OOOOOE+OO I
O.OOOOOEtOO + -9. 85988E- Ol I
Figure 9.2 Two P air of Complex Conjugate Eigenvalues and Eigenvectors
of t he Constant 4 x 4 Matrix in Equation ( 4)
Using t his information, we can write the real general solut ion of system ( 4)
as

(5) u(x) = ci[(cos,6x)v 1 - (sin,Bx)v2] + c2[(sin,6x)v 1 + (cos,6x)v2]