Mathematics

Differential Equations: Special Functions

A comprehensive study of the Legendre, Bessel, and Gauss (Hypergeometric) differential equations. This module covers series solutions, orthogonality, recurrence relations, and applications in mathematical physics.

Skill focus

Derive Legendre polynomials using Rodrigues' formula
Solve boundary value problems using Bessel functions
Apply orthogonality properties to function expansions
Manipulate Hypergeometric series to represent elementary functions

Lesson map

Progress through the sequence or jump to any lesson. Estimated total time: 360 minutes.

Progress: 0/3 (0%)

INTERACTIVECore120 min
Module 1: The Legendre Equation & Polynomials
Learning objectives
- Solve the Legendre ODE using power series
- Derive Legendre Polynomials via Rodrigues' Formula
- Master the Generating Function and Recurrence Relations
- Expand arbitrary functions in Legendre series
Guided steps
LegendreSpherical HarmonicsPotential Theory
INTERACTIVECore140 min
Module 2: The Bessel Equation & Functions
Learning objectives
- Solve Bessel's Equation of order k
- Differentiate between First Kind (J) and Second Kind (Y) functions
- Apply Recurrence Formulas for derivative manipulation
- Understand applications in cylindrical wave propagation
Guided steps
1. watch15 min
  Watch: Bessel Functions in Physics
  - Bessel Functions and the Vibrating Membrane
2. read30 min
  Solved Problem: Proving Identities
Prereq: legendre-equation-comprehensiveBesselCylindrical CoordinatesVibration Analysis
INTERACTIVECore100 min
Module 3: The Gauss (Hypergeometric) Equation
Learning objectives
- Recognize the general Gauss equation form
- Define the Hypergeometric Series F(a,b,c;x)
- Identify special cases (Legendre, Chebyshev, etc.)
- Test for convergence of the series
Guided steps
1. read25 min
  Solved Problem: Elementary Functions
Prereq: legendre-equation-comprehensiveHypergeometricSeries Solutions

Overview

Level: Advanced
Guided time: 600 min
Prerequisites: Calculus III (Multivariable)Differential Equations (ODEs)Infinite Series & Power Series
Keywords: #Legendre Polynomials#Bessel Functions#Hypergeometric Series#Frobenius Method#Orthogonality#Rodrigues Formula#Recurrence Relations

Lesson map

Progress through the sequence or jump to any lesson. Estimated total time: 360 minutes.

Progress: 0/3 (0%)

INTERACTIVECore120 min
Module 1: The Legendre Equation & Polynomials
Learning objectives
- Solve the Legendre ODE using power series
- Derive Legendre Polynomials via Rodrigues' Formula
- Master the Generating Function and Recurrence Relations
- Expand arbitrary functions in Legendre series
Guided steps
LegendreSpherical HarmonicsPotential Theory
INTERACTIVECore140 min
Module 2: The Bessel Equation & Functions
Learning objectives
- Solve Bessel's Equation of order k
- Differentiate between First Kind (J) and Second Kind (Y) functions
- Apply Recurrence Formulas for derivative manipulation
- Understand applications in cylindrical wave propagation
Guided steps
1. watch15 min
  Watch: Bessel Functions in Physics
  - Bessel Functions and the Vibrating Membrane
2. read30 min
  Solved Problem: Proving Identities
Prereq: legendre-equation-comprehensiveBesselCylindrical CoordinatesVibration Analysis
INTERACTIVECore100 min
Module 3: The Gauss (Hypergeometric) Equation
Learning objectives
- Recognize the general Gauss equation form
- Define the Hypergeometric Series F(a,b,c;x)
- Identify special cases (Legendre, Chebyshev, etc.)
- Test for convergence of the series
Guided steps
1. read25 min
  Solved Problem: Elementary Functions
Prereq: legendre-equation-comprehensiveHypergeometricSeries Solutions

Calculus Discovery

Connect limits, derivatives, and integrals with conceptual narratives and guided practice.

Advanced360 mins guided

#limits
#derivatives
#integrals
#rate of change

View lesson map ->

Mathematics

Number Theory and Cryptography

Investigate properties of integers, prime numbers, modular arithmetic, and applications to modern cryptography. Explore the mathematics behind internet security and encryption.

Advanced340 mins guided

#primes
#divisibility
#modular arithmetic
#GCD

View lesson map ->

Differential Equations: Special Functions

Skill focus

Lesson map

Module 1: The Legendre Equation & Polynomials

Module 2: The Bessel Equation & Functions

Module 3: The Gauss (Hypergeometric) Equation

Related topics

Calculus Discovery

Number Theory and Cryptography

Overview

Differential Equations: Special Functions

Skill focus

Lesson map

Module 1: The Legendre Equation & Polynomials

Module 2: The Bessel Equation & Functions

Module 3: The Gauss (Hypergeometric) Equation

Related topics

Calculus Discovery

Number Theory and Cryptography

Overview