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Problems In each of Problems 1 through 4, use Euler's formula to write the given expression in the form a + ib. 1. exp(2-3i)

Problems In each of Problems 1 through 4, use Euler's formula to write the given expression in the form a + ib. 1. exp(2-3i)

Advanced Engineering Mathematics
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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Note that if the real in the solution. Figure 3.3.3 shows the graph of two solutions of equation (28) conditions. In each case the solution is a pure oscillation with period 27/3 but whos and phase shift are determined by the initial conditions. Since there is no exponential solution (30), the amplitude of each oscillation remains constant in time. Problems In each of Problems 1 through 4, use Euler's formula to write the given expression in the form a + ib. 1. exp(2-3i) 2. eitt 3. 4. 2¹-1 In each of Problems 5 through 11, find the general solution of the given differential equation. 5. y" - 2y' + 2y = 0 6. y" - 2y' +6y=0 7. y" +2y + 2y = 0 8. y" + 6y +13y = 0 9. y"+2y' +1.25y = 0 10. 9y" +9y' - 4y = 0 11. y" +4y' +6.25y = 0 In each of Problems 12 through 15, find the solution of the given initial value problem. Sketch the graph of the solution and describe its behavior for increasing t. e²-(π/2)i G 12. y" + 4y = 0, y(0) = 0, y'(0) = 1 G 13. y"-2y' + 5y = 0, y(π/2) = 0, y'(π/2) = 2 G 14. y' G 15. y N 16. C a. F b. F N 17. a. b. N 18 a. b C V (
Transcribed Image Text:Note that if the real in the solution. Figure 3.3.3 shows the graph of two solutions of equation (28) conditions. In each case the solution is a pure oscillation with period 27/3 but whos and phase shift are determined by the initial conditions. Since there is no exponential solution (30), the amplitude of each oscillation remains constant in time. Problems In each of Problems 1 through 4, use Euler's formula to write the given expression in the form a + ib. 1. exp(2-3i) 2. eitt 3. 4. 2¹-1 In each of Problems 5 through 11, find the general solution of the given differential equation. 5. y" - 2y' + 2y = 0 6. y" - 2y' +6y=0 7. y" +2y + 2y = 0 8. y" + 6y +13y = 0 9. y"+2y' +1.25y = 0 10. 9y" +9y' - 4y = 0 11. y" +4y' +6.25y = 0 In each of Problems 12 through 15, find the solution of the given initial value problem. Sketch the graph of the solution and describe its behavior for increasing t. e²-(π/2)i G 12. y" + 4y = 0, y(0) = 0, y'(0) = 1 G 13. y"-2y' + 5y = 0, y(π/2) = 0, y'(π/2) = 2 G 14. y' G 15. y N 16. C a. F b. F N 17. a. b. N 18 a. b C V (
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