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7. y" + xy + 2y = 0, xo = 0 11

7. y" + xy + 2y = 0, xo = 0 11

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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Problems In each of Problems 1 through 11: a. Seek power series solutions of the given differential equation about the given point xo; find the recurrence relation that the coefficients must satisfy. b. Find the first four nonzero terms in each of two solutions y₁ and y2 (unless the series terminates sooner). c. By evaluating the Wronskian W[yı, y21(xo), show that y₁ and y2 form a fundamental set of solutions. d. If possible, find the general term in each solution. 1. y" - y = 0, 2. y" + 3y = 0, 3. y" - xy' - y = 0, 4. y" - xy' - y = 0, 5. y" +k²x²y = 0, 6. (1-x)y" + y = 0, 7. y" + xy' + 2y = 0, 8. xy" + y + xy = 0, 9. (3-x²) y" - 3xy' - y = 0, xo = 0 10. 2y" + xy' + 3y = 0, xo =0 11. 2y" + (x + 1) y' + 3y = 0, Xo = 2 In each of Problems 12 through 14: a. Find the first five nonzero terms in the solution of the given initial-value problem. Xo = = 0 12. 13. xo = 0 sec on. Xo = 0 Xo = 1 xo = 0, k a constant xo = 0 x0 = 0 xo = 1 ● y" - xy' - y = 0, y" + xy' + 2y = 0, Gb. Plot the four-term and the five-term approximations to the solution on the same axes. c. From the plot in part b, estimate the interval in which the four-term approximation is reasonably accurate. y(0) = 2, y'(0) = 1; see Problem 3 y(0) = 4, y'(0) = -1; see Problem 7 14. (1-x) y" + xy' - y = 0, y(0) = -3, y'(0) = 2 15. a. By making the change of variable x - 1 = t and assuming 17. of A of th 18. whe imp 19 In se the 5.3 SO
Transcribed Image Text:Problems In each of Problems 1 through 11: a. Seek power series solutions of the given differential equation about the given point xo; find the recurrence relation that the coefficients must satisfy. b. Find the first four nonzero terms in each of two solutions y₁ and y2 (unless the series terminates sooner). c. By evaluating the Wronskian W[yı, y21(xo), show that y₁ and y2 form a fundamental set of solutions. d. If possible, find the general term in each solution. 1. y" - y = 0, 2. y" + 3y = 0, 3. y" - xy' - y = 0, 4. y" - xy' - y = 0, 5. y" +k²x²y = 0, 6. (1-x)y" + y = 0, 7. y" + xy' + 2y = 0, 8. xy" + y + xy = 0, 9. (3-x²) y" - 3xy' - y = 0, xo = 0 10. 2y" + xy' + 3y = 0, xo =0 11. 2y" + (x + 1) y' + 3y = 0, Xo = 2 In each of Problems 12 through 14: a. Find the first five nonzero terms in the solution of the given initial-value problem. Xo = = 0 12. 13. xo = 0 sec on. Xo = 0 Xo = 1 xo = 0, k a constant xo = 0 x0 = 0 xo = 1 ● y" - xy' - y = 0, y" + xy' + 2y = 0, Gb. Plot the four-term and the five-term approximations to the solution on the same axes. c. From the plot in part b, estimate the interval in which the four-term approximation is reasonably accurate. y(0) = 2, y'(0) = 1; see Problem 3 y(0) = 4, y'(0) = -1; see Problem 7 14. (1-x) y" + xy' - y = 0, y(0) = -3, y'(0) = 2 15. a. By making the change of variable x - 1 = t and assuming 17. of A of th 18. whe imp 19 In se the 5.3 SO
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