e, we see they all lie on a circle of radius la¹/n and the angular separation between roots is 2π/n. Figure 3.20 shows the four roots of 4 + 16 = 0 found in Example 3. Im -√2+i√2 -√2-√2 √2 + √2 √2-√2 FIGURE 3.20 ede The four roots of 24+ 16 = 0 lie on a circle of radius 16¹/4 = 2 and have an angular separation of 2π/4 = π/2 radians. The roots occur in two complex conjugate pairs. Re Exercises 1-18: In each exercise, (a) Find the general solution of the differential equation. (b) If initial conditions are specified, solve the initial value problem. 2. y""+y" - y - y = 0 5. 16y(4) + 8y"+y=0 8. y(4) - y = 0 1. y" - 4y = 0 4. 16y(4) - 8y" + y = 0 7. y" - 2y" - y' + 2y = 0 10. 2y" - y" = 0 11. y" + y = 0 13. y(6) - y = 0 14. y(4) - y"+y'-y=0 15. y" + 2y" + y = 0, y(0) = 0, y'(0) = 0, y" (0) = 1 3. y"+y" + 4y + 4y = 0 6. y" - y = 0 9. y" +8y=0 12. y(4) + 2y" + y = 0
e, we see they all lie on a circle of radius la¹/n and the angular separation between roots is 2π/n. Figure 3.20 shows the four roots of 4 + 16 = 0 found in Example 3. Im -√2+i√2 -√2-√2 √2 + √2 √2-√2 FIGURE 3.20 ede The four roots of 24+ 16 = 0 lie on a circle of radius 16¹/4 = 2 and have an angular separation of 2π/4 = π/2 radians. The roots occur in two complex conjugate pairs. Re Exercises 1-18: In each exercise, (a) Find the general solution of the differential equation. (b) If initial conditions are specified, solve the initial value problem. 2. y""+y" - y - y = 0 5. 16y(4) + 8y"+y=0 8. y(4) - y = 0 1. y" - 4y = 0 4. 16y(4) - 8y" + y = 0 7. y" - 2y" - y' + 2y = 0 10. 2y" - y" = 0 11. y" + y = 0 13. y(6) - y = 0 14. y(4) - y"+y'-y=0 15. y" + 2y" + y = 0, y(0) = 0, y'(0) = 0, y" (0) = 1 3. y"+y" + 4y + 4y = 0 6. y" - y = 0 9. y" +8y=0 12. y(4) + 2y" + y = 0
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Please show all work and do all parts. Only do question 5.
Transcribed Image Text:e, we see they all
lie on a circle of radius lan and the angular separation between roots is 2π/n.
Figure 3.20 shows the four roots of λ + 16 = 0 found in Example 3.
-√2+i√2
-√2-√2
Im
√2+i√2
√2-√2
FIGURE 3.20
The four roots of 24+ 16 = 0 lie on a circle of radius 16¹/4 = 2 and have an
angular separation of 2π/4 = π/2 radians. The roots occur in two complex
conjugate pairs.
-Re
Exercises 1-18:
In each exercise,
(a) Find the general solution of the differential equation.
(b) If initial conditions are specified, solve the initial value problem.
2. y""+y" - y - y = 0
5. 16y(4) + 8y"+y=0
8. y(4) - y = 0
1. y" - 4y = 0
4. 16y(4) - 8y" + y = 0
7. y" - 2y" - y' + 2y = 0
10. 2y" - y" = 0
11. y" + y = 0
13. y(6) - y = 0
14. y(4) - y""+y'- y = 0
15. y" + 2y" + y = 0, y(0) = 0, y'(0) = 0, y" (0) = 1
3. y"+y" + 4y + 4y = 0
6. y" - y = 0
9. y" +8y=0
12. y(4) + 2y" + y = 0
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