Explanation Given: Let Q ( x ) be a polynomial of degree less than N . Approach: The Legendre equation, ( 1 − x 2 ) y ″ − 2 x y ′ + α ( α + 1 ) y = 0 Here, α is an arbitrary constant. Use the product rule and rewrite the Legendre equation as follows, [ ( 1 − x 2 ) y ′ ] + α ( α + 1 ) y = 0 Calculation: Substitute P N for y and N for α to satisfy the Legendre’s Polynomials. [ ( 1 − x 2 ) P ′ N ] + N ( N + 1 ) P N = 0 … ( 1 ) Substitute P M for y and M for α to satisfy the Legendre’s Polynomials. [ ( 1 − x 2 ) P ′ M ] + M ( M + 1 ) P M = 0 … ( 2 ) Multiply by P M in equation ( 1 ) and P N in equation ( 2 ) then subtract the multiplied new equations as follows. [ ( 1 − x 2 ) P ′ N ] P M − [ ( 1 − x 2 ) P ′ M ] P N + [ N ( N + 1 ) − M ( M + 1 ) ] P N P M = 0 [ ( 1 − x 2 ) ( P ′ N P M − P ′ M P N ) ] + [ N ( N + 1 ) − M ( M + 1 ) ] P N P M = 0 Integrate the above equation over the interval of [ − 1 , 1 ]

Differential Equations and Linear Algebra (4th Edition)

4th Edition

ISBN: 9780321964670

Author: Stephen W. Goode, Scott A. Annin

Publisher: PEARSON

See similar textbooks

Concept explainers

Minimization

In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in ter…

Maxima and Minima

The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.

Derivatives

A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which …

Concavity

In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the …

Videos

Question

Chapter 11.3, Problem 6P

To determine

To prove:

The equation ∫−11Q(x)PN(x)dx=0.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 11.3, Problem 5P

Chapter 11.3, Problem 7P

Students have asked these similar questions

e). n! (n - 1)!

Suppose you flip a fair two-sided coin four times and record the result. a). List the sample space of this experiment. That is, list all possible outcomes that could occur when flipping a fair two-sided coin four total times. Assume the two sides of the coin are Heads (H) and Tails (T).

e). n! (n - 1)!

Chapter 11 Solutions

Differential Equations and Linear Algebra (4th Edition)

Ch. 11.1 - True-False Review For Questions a-j, decide if the...Ch. 11.1 - Prob. 2TFR Ch. 11.1 - Prob. 3TFR Ch. 11.1 - Prob. 4TFR Ch. 11.1 - Prob. 5TFR Ch. 11.1 - True-False Review For Questions a-j, decide if the...Ch. 11.1 - Prob. 7TFR Ch. 11.1 - Prob. 8TFR Ch. 11.1 - Prob. 9TFR Ch. 11.1 - Prob. 10TFR

Knowledge Booster

$Algebra$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, algebra and related others by exploring similar questions and additional content below.