556 Ordinary Differential Equations
g. Let u1 = y, u2 = z, and u3 = z '. Then
13. a. ( -oo, oo)
c. (i) (-1,2)
f. (-oo, oo)u 1 (1) = -2, u2(l) = 3, u3(l) = 0
b. (i) (0, 7r /2) (ii) (7r /2, 37r /2)
(ii) (2,oo) d. (O,oo) e. (-00,00)
g. (-oo, oo)Chapter 8 LINEAR SYSTEMS OF FIRST-ORDER
DIFFERENTIAL EQUATIONS
Exercises 8.1 Matrices and Vectors
1. ( =! ~) 3. (-~~) 5. Cannot compute.
7 ( -D 9. Connot compute. 11 ( _; -n
- Cannot compute. 15. (-~~) 17. 4 19. -8 21. 1
- -1 25. -16 27. 2x^3 29. Yes 33. Linearly dependent
- Linearly dependent
Exercises 8.2 Eigenvalues and Eigenvectors
-1 = -1, ,\ (^2) =^2 ' Vl = (-1.78885 .44 721 4 ' ) V (^2) = ( -1.49071 1.49071)
3. >-1 = 1, >-2 = i,
(
0) ( .5 - .5i)
A3 = -i, V 1 = ~ ) V2 = -.5 7 .5i )