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3. y" - xy' - y = 0, xo=0-

3. y" - xy' - y = 0, xo=0-

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 yı and y2 (unless the series terminates sooner). c. By evaluating the Wronskian W[y₁, y2l(xo), show that yı and y2 form a fundamental set of solutions. d. If possible, find the general term in each solution. 1. y" - y = 0, xo = 0 2. y" + 3y' =0, xo = 0 3. y" - xy' - y = 0, y" - xy' - y = 0, y" +k²x²y = 0, (1-x)y"+y=0, y"+xy' + 2y = 0, xy" + y + xy = 0, ܝܕ ܚܕܐ ܚܕ ܘ ܗ 4. 5. 6. 7. 8. 9. 10. 11. xo = 0 xo = 1 xo = 0, k a constant xo = 0 xo = 0 Xo = 1 ~ (3-x²) y" - 3xy' - y = 0, xo = 0 2y" + xy' + 3y = 0, xo = 0 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. 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. 12. -13. 14. 15. a. By making the change of variable x - 1 = t and assuming that y has a Taylor series in powers of t, find two series solutions y"-xy'-y = 0, y(0) = 2, y'(0) = 1; see Problem 3 y" + xy' + 2y = 0, y(0) = 4, y'(0) = -1; see Problem 7 (1-x)y" + xy' - y = 0, y(0) = -3, y'(0) = 2 17. Show di of Airy's equa of the text. 18. The Her where A is ac important equa a. Find t about x = solutions. b. Observ or the othe polynomia 8, and 10. I multiplicati c. The He solution of coefficient 19. Consider th a. Show th G problem. b. Look for a power serie in x3 in this: In each of Proble series solution of thereby obtaining 5.2.4 (except that solution). 20. y" + xy G 21. (4-x2) G 22. y" + x²: G 23. (1-x) y
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 yı and y2 (unless the series terminates sooner). c. By evaluating the Wronskian W[y₁, y2l(xo), show that yı and y2 form a fundamental set of solutions. d. If possible, find the general term in each solution. 1. y" - y = 0, xo = 0 2. y" + 3y' =0, xo = 0 3. y" - xy' - y = 0, y" - xy' - y = 0, y" +k²x²y = 0, (1-x)y"+y=0, y"+xy' + 2y = 0, xy" + y + xy = 0, ܝܕ ܚܕܐ ܚܕ ܘ ܗ 4. 5. 6. 7. 8. 9. 10. 11. xo = 0 xo = 1 xo = 0, k a constant xo = 0 xo = 0 Xo = 1 ~ (3-x²) y" - 3xy' - y = 0, xo = 0 2y" + xy' + 3y = 0, xo = 0 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. 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. 12. -13. 14. 15. a. By making the change of variable x - 1 = t and assuming that y has a Taylor series in powers of t, find two series solutions y"-xy'-y = 0, y(0) = 2, y'(0) = 1; see Problem 3 y" + xy' + 2y = 0, y(0) = 4, y'(0) = -1; see Problem 7 (1-x)y" + xy' - y = 0, y(0) = -3, y'(0) = 2 17. Show di of Airy's equa of the text. 18. The Her where A is ac important equa a. Find t about x = solutions. b. Observ or the othe polynomia 8, and 10. I multiplicati c. The He solution of coefficient 19. Consider th a. Show th G problem. b. Look for a power serie in x3 in this: In each of Proble series solution of thereby obtaining 5.2.4 (except that solution). 20. y" + xy G 21. (4-x2) G 22. y" + x²: G 23. (1-x) y
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