Practice Legendre Functions (10.2) - Partial Differential Equations - Mathematics III (PDE, Probability & Statistics)
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Legendre Functions

Practice - Legendre Functions

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Practice Questions

Test your understanding with targeted questions

Question 1 Easy

What is the Legendre differential equation?

💡 Hint: Look for the equation that defines Legendre functions.

Question 2 Easy

What does 'n' represent in the Legendre equation?

💡 Hint: It's related to the degree of the polynomial.

4 more questions available

Interactive Quizzes

Quick quizzes to reinforce your learning

Question 1

What is the general form of the Legendre differential equation?

\\(\\frac{d}{dx}((1 - x^2)\\frac{dy}{dx}) + n(n + 1)y = 0\\)
\\(\\frac{d^2y}{dx^2} + p(x)\\frac{dy}{dx} + q(x)y = 0\\)

💡 Hint: Check for the terms involving 'n'.

Question 2

True or False: Legendre polynomials are orthogonal over the interval [-1, 1].

True
False

💡 Hint: Think about the definition of orthogonality.

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Challenge Problems

Push your limits with advanced challenges

Challenge 1 Hard

Prove the orthogonality of the first three Legendre polynomials over the interval [-1, 1].

💡 Hint: Set up the integrals accordingly.

Challenge 2 Hard

Using Legendre polynomials, derive a series solution for a given potential in a gravitational field based on spherical symmetry.

💡 Hint: Think about how they approximate potentials.

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Reference links

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