Differential Equations

5.

Series Solutions of Linear Equations

5.1.

Power Series Review

5.1.1.

Definition and Notation

5.1.2.

Convergence Tests

5.1.2.1.

5.1.2.2.

5.1.3.

Radius of Convergence

5.1.4.

Interval of Convergence

5.1.5.

Operations on Power Series

5.1.5.1.

Term-by-Term Differentiation

5.1.5.2.

Term-by-Term Integration

5.1.5.3.

Series Multiplication

5.2.

Solutions About Ordinary Points

5.2.1.

Definition of Ordinary Points

5.2.2.

Power Series Solution Method

5.2.2.1.

Assumption of Series Form

5.2.2.2.

Coefficient Determination

5.2.3.

Recurrence Relations

5.2.3.1.

Formation and Solution

5.2.3.2.

Pattern Recognition

5.2.4.

Convergence of Series Solutions

5.2.5.

Examples and Applications

5.3.

Solutions About Singular Points

5.3.1.

Classification of Singular Points

5.3.1.1.

Regular Singular Points

5.3.1.2.

Irregular Singular Points

5.3.2.

Method of Frobenius

5.3.2.1.

Frobenius Series Form

5.3.2.2.

Indicial Equation

5.3.2.2.1.

Formation and Solution

5.3.2.3.

Case Analysis by Root Differences

5.3.2.3.1.

Distinct Roots Not Differing by Integer

5.3.2.3.2.

5.3.2.3.3.

Roots Differing by Positive Integer

5.3.3.

Second Solution Construction

5.3.3.1.

Logarithmic Terms

5.3.3.2.

Reduction of Order Application

5.4.

Special Functions and Their Properties

5.4.1.

Bessel's Differential Equation

5.4.1.1.

5.4.1.2.

Parameter Significance

5.4.2.

Bessel Functions of the First Kind

5.4.2.1.

Definition and Properties

5.4.2.2.

Recurrence Relations

5.4.2.3.

Orthogonality Properties

5.4.3.

Bessel Functions of the Second Kind

5.4.3.1.

Definition and Properties

5.4.3.2.

Singular Behavior

5.4.4.

Modified Bessel Functions

5.4.5.

Legendre's Differential Equation

5.4.5.1.

5.4.5.2.

Parameter Significance

5.4.6.

Legendre Polynomials

5.4.6.1.

Definition and Properties

5.4.6.2.

Orthogonality on [-1,1]

5.4.6.3.

Generating Function

5.4.6.4.

Recurrence Relations

5.4.7.

Applications of Special Functions

5.4.7.1.

Physical Problems

5.4.7.2.

Boundary Value Problems

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6. Systems of Linear First-Order Differential Equations