For Problems 14-16, show that the given functions inCº[-1, 1] are orthogonal, and use them to construct an or-thonormal set of functions in Cº[-1, 1]. 15. f1(x) = 1, ƒ2(x) = x, f3(x) = ½(3x² − 1).= -[Note: These are the Legendre polynomials that ariseas solutions of the Legendre differential equation(1-x²)y" - 2xy' + n(n+1)y = 0,when n = 0, 1, 2, respectively.]

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Answered: - 5. f1(x) = 1, ƒ2(x) = x, ƒ3(x) = (3x²…

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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2 1. Consider a polynomial function f(x) = co + C₁ x + c₂x² +...+ cd-1xd-1 + caxd. A. Applying the addition rule, the constant multiple rule, and the power rule for derivatives, we can compute that ƒ'(x) = c₁ + 2c2x + … + (d − 1)cd-1xd-2 + (d)caxd-¹. Explain. B. Following the general outline from Part A, compute the derivatives of each of the following functions: g(x) = 5x³ 10x² + 7x - 11. p(x) = −7x² – 3x² + πx √17. O O 2. Consider the function f(x) = (2x³ – 5x + 1) (3x² − 4). A. Distribute out the product of the two polynomials multiplied in the expression of f and simplify your result to find an equivalent expression for f(x).
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2.
Use the Gamma function
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- 5. f1(x) = 1, ƒ2(x) = x, ƒ3(x) = (3x² − 1). [Note: These are the Legendre polynomials that arise as solutions of the Legendre differential equation (1-x²)y" - 2xy' + n(n + 1)y = 0, when n = 0, 1, 2, respectively.]