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Get Full Access to Differential Equations - 4 Edition - Chapter 11.3 - Problem 10
Get Full Access to Differential Equations - 4 Edition - Chapter 11.3 - Problem 10

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# When suitably normalized, the polynomial solutions to Equation (11.3.13) are called the ISBN: 9780321964670 380

## Solution for problem 10 Chapter 11.3

Differential Equations | 4th Edition

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Problem 10

When suitably normalized, the polynomial solutions to Equation (11.3.13) are called the Hermite polynomials, and are denoted by HN (x). (a) Use Equation (11.3.13) to show that HN (x) satisfies (ex2 H N ) + 2N ex2 HN = 0. (11.3.14) [Hint: Replace with N in Equation (11.3.13) and multiply the resulting equation by ex2 .] (b) Use Equation (11.3.14) to prove that the Hermite polynomials satisfy ex2 HN (x)HM (x)dx = 0, M = N. (11.3.15) [Hint: Follow the steps taken in proving orthogonality of the Legendre polynomials. You will need to recall that lim x ex2 p(x) = 0, for any polynomial p.] (c) Let p(x) be a polynomial of degree N. Then we can write p(x) = N k=1 ak Hk (x). (11.3.16) Given that ex2 H2 N (x)dx = 2N N! , use (11.3.15) to prove that the constants in (11.3.16) are given by a j = 1 2 j j! ex2 Hj(x)p(x)dx.

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##### ISBN: 9780321964670

This textbook survival guide was created for the textbook: Differential Equations, edition: 4. The answer to “When suitably normalized, the polynomial solutions to Equation (11.3.13) are called the Hermite polynomials, and are denoted by HN (x). (a) Use Equation (11.3.13) to show that HN (x) satisfies (ex2 H N ) + 2N ex2 HN = 0. (11.3.14) [Hint: Replace with N in Equation (11.3.13) and multiply the resulting equation by ex2 .] (b) Use Equation (11.3.14) to prove that the Hermite polynomials satisfy ex2 HN (x)HM (x)dx = 0, M = N. (11.3.15) [Hint: Follow the steps taken in proving orthogonality of the Legendre polynomials. You will need to recall that lim x ex2 p(x) = 0, for any polynomial p.] (c) Let p(x) be a polynomial of degree N. Then we can write p(x) = N k=1 ak Hk (x). (11.3.16) Given that ex2 H2 N (x)dx = 2N N! , use (11.3.15) to prove that the constants in (11.3.16) are given by a j = 1 2 j j! ex2 Hj(x)p(x)dx.” is broken down into a number of easy to follow steps, and 157 words. Since the solution to 10 from 11.3 chapter was answered, more than 217 students have viewed the full step-by-step answer. This full solution covers the following key subjects: . This expansive textbook survival guide covers 91 chapters, and 2967 solutions. Differential Equations was written by and is associated to the ISBN: 9780321964670. The full step-by-step solution to problem: 10 from chapter: 11.3 was answered by , our top Math solution expert on 03/13/18, 06:45PM.

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