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5. y" +k²x²y = 0, xo = 0, k a constant 6 (1 x)!!

5. y" +k²x²y = 0, xo = 0, k a constant 6 (1 x)!!

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). id qayoq c. By evaluating the Wronskian W[y1, y21(xo), show that yı and Y/2 form a fundamental set of solutions. d. If possible, find the general term in each solution. 1. y - y = 0, xo = 0 xo = 0 2. y" + 3y' =0, xo = 0 xo = 1 3. y" -xy' - y = 0, 4. y" - xy' - y = 0, 5. y" +k²x²y = 0, (1-x)y" + y = 0, 6. 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. G b. 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 - xo = 0, k a constant xo = 0 xo = 0 xo = 1 195 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" + xy' +2y = 0, (1-x)y" + xy' - y = 0, y(0) = -3, y'(0) = 2 17. Sho of Airy's of the text 18. The where i importante a. Fi about solutic b. Ob or the polyno 8, and multipl c. The solution coeffici 19. Consid a. Sho problem b. Look a power in x3 in t In each of Pr series solution thereby obtain 5.2.4 (except t solution). G 20. y" + 21. (4- G G 22. y" + G 23.
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). id qayoq c. By evaluating the Wronskian W[y1, y21(xo), show that yı and Y/2 form a fundamental set of solutions. d. If possible, find the general term in each solution. 1. y - y = 0, xo = 0 xo = 0 2. y" + 3y' =0, xo = 0 xo = 1 3. y" -xy' - y = 0, 4. y" - xy' - y = 0, 5. y" +k²x²y = 0, (1-x)y" + y = 0, 6. 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. G b. 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 - xo = 0, k a constant xo = 0 xo = 0 xo = 1 195 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" + xy' +2y = 0, (1-x)y" + xy' - y = 0, y(0) = -3, y'(0) = 2 17. Sho of Airy's of the text 18. The where i importante a. Fi about solutic b. Ob or the polyno 8, and multipl c. The solution coeffici 19. Consid a. Sho problem b. Look a power in x3 in t In each of Pr series solution thereby obtain 5.2.4 (except t solution). G 20. y" + 21. (4- G G 22. y" + G 23.
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