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-13. y"+xy' +2y = 0, y(0) = 4, y'(0) = -1; 14. (1-x)y"+ry'-x-0 (0) 100%

-13. y"+xy' +2y = 0, y(0) = 4, y'(0) = -1; 14. (1-x)y"+ry'-x-0 (0) 100%

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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12. -13. 13. 14. 15. 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[y1, y21(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, y" + 3y' = 0, 3. y"-xy' - y = 0, 4. y" - xy' - y = 0, 5. y" +k²x²y = 0, 6. (1-x)y" + y = 0, 7. 8. a pot of 2. 9. Xo = 0 Xo = 0 xo = 0. Xo = 1 xo = 0, k a constant Xo = 0 хо Xo = 0 xo=1 y"+xy' + 2y = 0, xy" + y + xy = 0, (3-x²)y" - 3xy' - y = 0, Xo = 0 хо xo = 0 10. 2y" + xy' + 3y = 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. 8001 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 ni boazm y" + (x − 1)²y' + (x² - 1)y=0 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). d 17. Show directly, using the ra of Airy's equation about x = 0 of the text. 18. The Hermite Equation. y" - 2xy + y = where is a constant, is know important equation in mathemat a. Find the first four nor about x = 0 and show solutions. F b. Observe that if is a or the other of the series polynomial. Find the poly 8, and 10. Note that each multiplicative constant. c. The Hermite polynomi solution of the Hermite e coefficient of x" is 2". Fine 19. Consider the initial-value a. Show that y sin x problem. b. Look for a solution of ti a power series about x = ( in x3 in this series. In each of Problems 20 throug series solution of the given in thereby obtaining graphs analog 5.2.4 (except that we do not kn solution). 20. y" + xy' + 2y = 0, G 21. (4-x²) y" + 2y = G 22. y" + x²y = 0, y(0) G 23. (1-x)y" + xy' -23 5Charles Hermite (1822-1901) w algebraist. An inspiring teacher, he w and the Sorbonne. He introduced the 1873 that e is a transcendental numbe equation with rational coefficients). H matrices (see Section 7.3), some of w
Transcribed Image Text:12. -13. 13. 14. 15. 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[y1, y21(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, y" + 3y' = 0, 3. y"-xy' - y = 0, 4. y" - xy' - y = 0, 5. y" +k²x²y = 0, 6. (1-x)y" + y = 0, 7. 8. a pot of 2. 9. Xo = 0 Xo = 0 xo = 0. Xo = 1 xo = 0, k a constant Xo = 0 хо Xo = 0 xo=1 y"+xy' + 2y = 0, xy" + y + xy = 0, (3-x²)y" - 3xy' - y = 0, Xo = 0 хо xo = 0 10. 2y" + xy' + 3y = 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. 8001 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 ni boazm y" + (x − 1)²y' + (x² - 1)y=0 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). d 17. Show directly, using the ra of Airy's equation about x = 0 of the text. 18. The Hermite Equation. y" - 2xy + y = where is a constant, is know important equation in mathemat a. Find the first four nor about x = 0 and show solutions. F b. Observe that if is a or the other of the series polynomial. Find the poly 8, and 10. Note that each multiplicative constant. c. The Hermite polynomi solution of the Hermite e coefficient of x" is 2". Fine 19. Consider the initial-value a. Show that y sin x problem. b. Look for a solution of ti a power series about x = ( in x3 in this series. In each of Problems 20 throug series solution of the given in thereby obtaining graphs analog 5.2.4 (except that we do not kn solution). 20. y" + xy' + 2y = 0, G 21. (4-x²) y" + 2y = G 22. y" + x²y = 0, y(0) G 23. (1-x)y" + xy' -23 5Charles Hermite (1822-1901) w algebraist. An inspiring teacher, he w and the Sorbonne. He introduced the 1873 that e is a transcendental numbe equation with rational coefficients). H matrices (see Section 7.3), some of w
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