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7. y"+xy' +2y = 0, 11 0 Xo = 0

7. y"+xy' +2y = 0, 11 0 Xo = 0

Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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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, y21(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, xo = 0 xo = 1 4. y"-xy' - y = 0, 5. y"+k²x²y = 0, 6. (1-x)y"+y=0, 7. y" + xy' + 2y = 0, 8. xy"+y' + xy = 0, xo = 0, k a constant xo = 0 xo = 0 xo = 1 9. (3-x²)y" - 3xy' - y = 0, Xo = 0 xo=0 10. 2y" + xy' + 3y = 0, 11. 2y" + (x + 1) y' + 3y = 0, xo = 2 xo = 0 In each of Problems 12 through 14: a. Find the first five nonzero terms in the solution of the given initial-value problem. ons Gb. Plot the four-term and the five-term approximations to the solution on the same axes. mount c. From the plot in part b, estimate the interval in which the four-term approximation is reasonably accurate. 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 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 y" + (x - 1)² y' + (x² - 1) y = 0 17. Show directly, using the ratio test, that the two series solutions of Airy's equation about x = 0 converge for all x; see equation (20) of the text. 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). 18. The Hermite Equation. The equation y" - 2xy' + Ay = 0, -∞0 < x < ∞, where A is a constant, is known as the Hermites equation. It is an important equation in mathematical physics. OST 19. Consider the initial-value problem y'=√1-y2, y(0) = 0. a. Show that y = sinx is the solution of this initial-value problem. b. Look for a solution of the initial-value problem in the form of a power series about x = 0. Find the coefficients up to the term in x3 in this series. a. Find the first four nonzero terms in each of two solutions about x = 0 and show that they form a fundamental set of solutions. b. Observe that if X is a nonnegative even integer, then one or the other of the series solutions terminates and becomes a polynomial. Find the polynomial solutions for λ = 0, 2, 4, 6, 8, and 10. Note that each polynomial is determined only up to a multiplicative constant. c. The Hermite polynomial H₂(x) is defined as the polynomial solution of the Hermite equation with λ = 2n for which the coefficient of x" is 2". Find Ho(x), H₁(x), .. ..., H5(x). REIS In each of Problems 20 through 23, plot several partial sums in a series solution of the given initial-value problem about x = 0, thereby obtaining graphs analogous to those in Figures 5.2.1 through 5.2.4 (except that we do not know an explicit formula for the actual solution). y"+xy' + 2y = 0, y(0) = 0, y'(0) = 1; see Problem 7 (4-x2) y" + 2y = 0, y(0) = 0, y'(0) = G 22. y" + x²y = 0, y(0) = 1, y'(0) = 0; y(0) = 1, y'(0) = 0; see Problem 5 y(0) = 0, y'(0) = 1 G 23. (1-x) y" + xy' - 2y = 0, siem nessWszitemsile.na att 02 G 20. G 21. no Charles Hermite (1822-1901) was an influential French analyst and algebraist. An inspiring teacher, he was professor at the École Polytechnique and the Sorbonne. He introduced the Hermite functions in 1864 and showed in 1873 that e is a transcendental number (that is, e is not a root of any polynomial equation with rational coefficients). His name is also associated with Hermitian matrices (see Section 7.3), some of whose properties he discovered.
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, y21(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, xo = 0 xo = 1 4. y"-xy' - y = 0, 5. y"+k²x²y = 0, 6. (1-x)y"+y=0, 7. y" + xy' + 2y = 0, 8. xy"+y' + xy = 0, xo = 0, k a constant xo = 0 xo = 0 xo = 1 9. (3-x²)y" - 3xy' - y = 0, Xo = 0 xo=0 10. 2y" + xy' + 3y = 0, 11. 2y" + (x + 1) y' + 3y = 0, xo = 2 xo = 0 In each of Problems 12 through 14: a. Find the first five nonzero terms in the solution of the given initial-value problem. ons Gb. Plot the four-term and the five-term approximations to the solution on the same axes. mount c. From the plot in part b, estimate the interval in which the four-term approximation is reasonably accurate. 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 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 y" + (x - 1)² y' + (x² - 1) y = 0 17. Show directly, using the ratio test, that the two series solutions of Airy's equation about x = 0 converge for all x; see equation (20) of the text. 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). 18. The Hermite Equation. The equation y" - 2xy' + Ay = 0, -∞0 < x < ∞, where A is a constant, is known as the Hermites equation. It is an important equation in mathematical physics. OST 19. Consider the initial-value problem y'=√1-y2, y(0) = 0. a. Show that y = sinx is the solution of this initial-value problem. b. Look for a solution of the initial-value problem in the form of a power series about x = 0. Find the coefficients up to the term in x3 in this series. a. Find the first four nonzero terms in each of two solutions about x = 0 and show that they form a fundamental set of solutions. b. Observe that if X is a nonnegative even integer, then one or the other of the series solutions terminates and becomes a polynomial. Find the polynomial solutions for λ = 0, 2, 4, 6, 8, and 10. Note that each polynomial is determined only up to a multiplicative constant. c. The Hermite polynomial H₂(x) is defined as the polynomial solution of the Hermite equation with λ = 2n for which the coefficient of x" is 2". Find Ho(x), H₁(x), .. ..., H5(x). REIS In each of Problems 20 through 23, plot several partial sums in a series solution of the given initial-value problem about x = 0, thereby obtaining graphs analogous to those in Figures 5.2.1 through 5.2.4 (except that we do not know an explicit formula for the actual solution). y"+xy' + 2y = 0, y(0) = 0, y'(0) = 1; see Problem 7 (4-x2) y" + 2y = 0, y(0) = 0, y'(0) = G 22. y" + x²y = 0, y(0) = 1, y'(0) = 0; y(0) = 1, y'(0) = 0; see Problem 5 y(0) = 0, y'(0) = 1 G 23. (1-x) y" + xy' - 2y = 0, siem nessWszitemsile.na att 02 G 20. G 21. no Charles Hermite (1822-1901) was an influential French analyst and algebraist. An inspiring teacher, he was professor at the École Polytechnique and the Sorbonne. He introduced the Hermite functions in 1864 and showed in 1873 that e is a transcendental number (that is, e is not a root of any polynomial equation with rational coefficients). His name is also associated with Hermitian matrices (see Section 7.3), some of whose properties he discovered.
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