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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). 1. 2. 3. 4. 5. 6. 7. 8. 9. c. By evaluating the Wronskian W[y1, y21(xo), show that yı and form a fundamental set of solutions. 2 d. If possible, find the general term in each solution. y" - y = 0, y" + 3y = 0, y" -xy' - y = 0, y" - xy' - y = 0, y" +k²x²y = 0, Xo = 0 xo = 0 xo = 0. xo = 1 xo = 0, k a constant xo = 0 Xo = 0 Xo = 1 (1-x)y"+y=0, y"+xy' + 2y = 0, xy" + y + xy = 0, (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. SPLIT 12. y" - xy' - y = 0, y(0) = 2, y'(0) = 1; see Problem 3 -13. y" + xy + 2y = 0, y(0) = 4, y'(0) = -1; see Problem 7 14. (1-x) y" + xy' - y = 0, y(0) = -3, y'(0) = 2 = t and assuming 15. a. By making the change of variable x - 1 o Taylor series in powers of t, find two series solutions 17. Show of Airy's eq of the text. 18. The H where is a important eq a. Finc about x solution. b. Obse or the o polynom 8, and 10 multiplic c. The E solution coefficier 19. Consider a. Show problem. b. Look a power s in x3 in th In each of Pro series solution thereby obtaini 5.2.4 (except th solution). G 20. y" + G 21. (4- G 22. !!

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). 1. 2. 3. 4. 5. 6. 7. 8. 9. c. By evaluating the Wronskian W[y1, y21(xo), show that yı and form a fundamental set of solutions. 2 d. If possible, find the general term in each solution. y" - y = 0, y" + 3y = 0, y" -xy' - y = 0, y" - xy' - y = 0, y" +k²x²y = 0, Xo = 0 xo = 0 xo = 0. xo = 1 xo = 0, k a constant xo = 0 Xo = 0 Xo = 1 (1-x)y"+y=0, y"+xy' + 2y = 0, xy" + y + xy = 0, (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. SPLIT 12. y" - xy' - y = 0, y(0) = 2, y'(0) = 1; see Problem 3 -13. y" + xy + 2y = 0, y(0) = 4, y'(0) = -1; see Problem 7 14. (1-x) y" + xy' - y = 0, y(0) = -3, y'(0) = 2 = t and assuming 15. a. By making the change of variable x - 1 o Taylor series in powers of t, find two series solutions 17. Show of Airy's eq of the text. 18. The H where is a important eq a. Finc about x solution. b. Obse or the o polynom 8, and 10 multiplic c. The E solution coefficier 19. Consider a. Show problem. b. Look a power s in x3 in th In each of Pro series solution thereby obtaini 5.2.4 (except th solution). G 20. y" + G 21. (4- G 22. !!

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. xo = 0 1. y" - y = 0, 2. y" + 3y = 0, 3. y" -xy' - y = 0, 4. y"-xy'- y = 0, 5. 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. 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. xo = 0 xo = 0 xo = 1 xo = 0, ka constant xo = 0 Xo = 0 - xo = 1 ers of x - 1. 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 of 1- 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 y" + (x − 1)²y' + (x² - 1) y = 0 17. Show direc of Airy's equation of the text. 18. The Hermit y" - where λ is a cons important equation a. Find the about x solutions. b. Observe t or the other c polynomial. F 8, and 10. Not multiplicative c. The Hermi solution of the coefficient of x = ( 19. Consider the i a. Show that problem. b. Look for a s a power series a in x3 in this seri In each of Problems series solution of th thereby obtaining gra 5.2.4 (except that we solution). G 20. y" + xy + G 21. (4-x²) y' G 22. y" + x²y = G 23. (1-x) y" - 5 Charles Hermite (1
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. xo = 0 1. y" - y = 0, 2. y" + 3y = 0, 3. y" -xy' - y = 0, 4. y"-xy'- y = 0, 5. 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. 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. xo = 0 xo = 0 xo = 1 xo = 0, ka constant xo = 0 Xo = 0 - xo = 1 ers of x - 1. 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 of 1- 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 y" + (x − 1)²y' + (x² - 1) y = 0 17. Show direc of Airy's equation of the text. 18. The Hermit y" - where λ is a cons important equation a. Find the about x solutions. b. Observe t or the other c polynomial. F 8, and 10. Not multiplicative c. The Hermi solution of the coefficient of x = ( 19. Consider the i a. Show that problem. b. Look for a s a power series a in x3 in this seri In each of Problems series solution of th thereby obtaining gra 5.2.4 (except that we solution). G 20. y" + xy + G 21. (4-x²) y' G 22. y" + x²y = G 23. (1-x) y" - 5 Charles Hermite (1
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