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-13. y"+xy' +2y = 0, y(0) = 4, y'(0) = -1; see Problem 7 14 vy! 0 (0) --3 v'(0) = 2 11 -

-13. y"+xy' +2y = 0, y(0) = 4, y'(0) = -1; see Problem 7 14 vy! 0 (0) --3 v'(0) = 2 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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In each of Problems I through 11: a. Seek power series solutions of the given differential 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, y21(xo), show that yı and y2 form a fundamental set of solutions, d. If possible, find the general term in each solution. y" - y = 0, y" + 3y' = 0, xo = 0 xo = 1 y" - xy' - y = 0, y" - xy' - y = 0, 5. y" +k²x²y = 0, 6. (1-x)y"+y=0, y" + xy' + 2y = 0, xo = 0, k a constant xo = 0 хо xo = 0 xy" + y + xy = 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 1. 2. 3. 4. ܝܕ ܝܕ ܚܕ ܝܕ ܬܐ ܗ ܝ 7. 8. 9. 10. 11. Xo = 0 12. 13. 14. 15. Xo = 0 18. The In each of Problems 12 through 14: a. Find the first five nonzero terms in the solution of the given initial-value problem. where Xi important a. I abou solu b. y" + (x − 1)²y' + (x² - 1) y = 0 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" - 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 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 of or pol 8, mu C. SC CO 19. in powers of x - 1. b. Show that you obtain the same result by assuming that y has a Taylor series in powers of x - 1 and also expressing the coefficient x² - 1 in powers of x - 1. 16. Prove equation (10). a F In ea serie ther 5.2. solu G G G SC al a 1
Transcribed Image Text:In each of Problems I through 11: a. Seek power series solutions of the given differential 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, y21(xo), show that yı and y2 form a fundamental set of solutions, d. If possible, find the general term in each solution. y" - y = 0, y" + 3y' = 0, xo = 0 xo = 1 y" - xy' - y = 0, y" - xy' - y = 0, 5. y" +k²x²y = 0, 6. (1-x)y"+y=0, y" + xy' + 2y = 0, xo = 0, k a constant xo = 0 хо xo = 0 xy" + y + xy = 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 1. 2. 3. 4. ܝܕ ܝܕ ܚܕ ܝܕ ܬܐ ܗ ܝ 7. 8. 9. 10. 11. Xo = 0 12. 13. 14. 15. Xo = 0 18. The In each of Problems 12 through 14: a. Find the first five nonzero terms in the solution of the given initial-value problem. where Xi important a. I abou solu b. y" + (x − 1)²y' + (x² - 1) y = 0 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" - 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 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 of or pol 8, mu C. SC CO 19. in powers of x - 1. b. Show that you obtain the same result by assuming that y has a Taylor series in powers of x - 1 and also expressing the coefficient x² - 1 in powers of x - 1. 16. Prove equation (10). a F In ea serie ther 5.2. solu G G G SC al a 1
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