3 ProblemsIn each of Problems 1 through 11:a. Seek power series solutions of the given differential equationabout the given point xo; find the recurrence relation that thecoefficients must satisfy.b. Find the first four nonzero terms in each of two solutions yıand y2 (unless the series terminates sooner).c. By evaluating the Wronskian W[y1, y21(xo), show that yıand y2 form a fundamental set of solutions.d. If possible, find the general term in each solution.Xo = 01.y" - y = 0,2.y" + 3y = 0,3.y" - xy' - y = 0,xo = 0xo = 14. 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 constantxo = 0xo = 0xo = 19.(3-x²)y" - 3xy' - y = 0, Xo = 0xo=010. 2y" + xy' + 3y = 0,11. 2y" + (x + 1) y' + 3y = 0, xo = 2xo = 0In each of Problems 12 through 14:a. Find the first five nonzero terms in the solution of the giveninitial-value problem.onsGb. Plot the four-term and the five-term approximations to thesolution on the same axes.mountc. From the plot in part b, estimate the interval in which thefour-term approximation is reasonably accurate.12. y" - xy' - y = 0, y(0) = 2, y'(0) = 1;see Problem 313. y" + xy' + 2y = 0, y(0) = 4, y'(0) = -1; see Problem 714. (1-x)y" + xy' - y = 0, y(0) = -3, y'(0) = 215. a. By making the change of variable x - 1 = t and assumingthat y has a Taylor series in powers of t, find two series solutionsofy" + (x - 1)² y' + (x² - 1) y = 017. Show directly, using the ratio test, that the two series solutionsof 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 yhas a Taylor series in powers of x - 1 and also expressing thecoefficient x² - 1 in powers of x - 1.16. Prove equation (10).18. The Hermite Equation. The equationy" - 2xy' + Ay = 0, -∞0 < x < ∞,where A is a constant, is known as the Hermites equation. It is animportant 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-valueproblem.b. Look for a solution of the initial-value problem in the form ofa power series about x = 0. Find the coefficients up to the termin x3 in this series.a. Find the first four nonzero terms in each of two solutionsabout x = 0 and show that they form a fundamental set ofsolutions.b. Observe that if X is a nonnegative even integer, then oneor the other of the series solutions terminates and becomes apolynomial. Find the polynomial solutions for λ = 0, 2, 4, 6,8, and 10. Note that each polynomial is determined only up to amultiplicative constant.c. The Hermite polynomial H₂(x) is defined as the polynomialsolution of the Hermite equation with λ = 2n for which thecoefficient of x" is 2". Find Ho(x), H₁(x), ....., H5(x).REISIn each of Problems 20 through 23, plot several partial sums in aseries solution of the given initial-value problem about x = 0,thereby obtaining graphs analogous to those in Figures 5.2.1 through5.2.4 (except that we do not know an explicit formula for the actualsolution).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 5y(0) = 0, y'(0) = 1G 23.(1-x) y" + xy' - 2y = 0,siem nessWszitemsile.na att 02G 20.G 21.noCharles Hermite (1822-1901) was an influential French analyst andalgebraist. An inspiring teacher, he was professor at the École Polytechniqueand the Sorbonne. He introduced the Hermite functions in 1864 and showed in1873 that e is a transcendental number (that is, e is not a root of any polynomialequation with rational coefficients). His name is also associated with Hermitianmatrices (see Section 7.3), some of whose properties he discovered.

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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 y and y2 (unless the series terminates sooner). c. By evaluating the Wronskian W[y1, y21(xo), show that yı and 3/2 form a fundamental set of solutions. d. If possible, find the general term in each solution. 1. y" - y = 0, 2. y" + 3y = 0, Xo = 0 3. y"-xy' - y = 0, xo = 0 x0=0

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 y and y2 (unless the series terminates sooner). c. By evaluating the Wronskian W[y1, y21(xo), show that yı and 3/2 form a fundamental set of solutions. d. If possible, find the general term in each solution. 1. y" - y = 0, 2. y" + 3y = 0, Xo = 0 3. y"-xy' - y = 0, xo = 0 x0=0

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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