## Lecture Notes

### Differential Equations

Much part of these lecture notes came from (Ordinary, Partial) Differential Equations courses I taught.

These lecture notes should be accessible by undergraduate students of mathematics or physics who have taken Calculus, Multi-Variable Calculus, and preferrably also Linear Algebra.

#### Lecture Notes

#### Ordinary Differential Equations

#### Partial Differential Equations

#### Problem Sets

#### Ordinary Differential Equations

#### Partial Differential Equations

#### References and Further Reading

Click on linked topics to view lecture notes. Lecture notes on Partial Differential Equations follow lecture notes on Ordinary Differential Equations. To go to lecture notes on Partial Differential Equations directly click here.

First-Order Differential Equations: Separable Case

First-Order Linear Differential Equations

Homogeneous Differential Equations

Bernoulli's Differential Equations

Exact Differential Equations 1

Exact Differential Equations 2

Harmonic Motion: Undamped

Harmonic Motion: Damped

Second-Order Linear Differential Equations and Linear Algebra

Non-Homogeneous Second-Order Differential Equations: The Method of Undetermined Coefficients

Non-Homogeneous Second-Order Differential Equations: The Method of Variation of Parameters

The Laplace Transform: Introduction

The Laplace Transform: Transforms of Derivatives

The Laplace Transform: The Inverse Transforms

The Laplace Transform: Solving Differential Equations

The Laplace Transform: Further Properties

The Laplace Transform: Convolution

The Laplace Transform: Differential Equations with Variable Coefficients

The Laplace Transform: Forced Vibration without Damping and Resonance

1-Dimensional Heat Initial Boundary Value Problems 1: Separation of Variables

1-Dimensional Heat Initial Boundary Value Problems 2: Sturm-Liouville Problems and Orthogonal Functions

1-Dimensional Heat Initial Boundary Value Problems 3: An Example of Heat IBVP with Mixed Boundary Conditions (Insulated and Specified Flux)

SolvingHeat Equation with Non-Homogeneous BCs 1: Time-Independent BCs

Solving Heat Equation with Non-Homogeneous BCs 2: Time-Dependet BCs

The Semi-Homogeneous Heat Problem

Modeling a Vibrating Drumhead I

Modeling a Vibrating Drumhead II

Modeling a Vibrating Drumhead III

Helmholtz Equation

Bessel Functions of the First Kind \(J_n(x)\) I: Generating Function, Recurrence Relation, Bessel's Equaiton

Cylindrical Resonant Cavity

Bessel Functions of the First Kind \(J_n(x)\) II: Orthogonality

Neumann Functions, Bessel Functions of the Second Kind \(N_\nu(X)\)

Spherical Bessel Functions

Legendre Functions I: A Physical Origin of Legendre Functions

Legendre Functions II: Reccurence Relations and Special Properties

Legendre Functions III:Special Values, Parity, Orthogonality

Self-Adjoint Differential Equations I

Self-Adjoint Differential Equations II: Hermitian Operators

Self-Adjoint Differential Equations III: Real Eigenvalues, Gram-Schmidt Orthogonalization

Heat Equation and SchrÃ¶dinger Equation

Click on the following links for problems. Problem sets on Partial Differential Equations follow problem sets on Ordinary Differential Equations. To go to problem sets on Partial Differential Equations directly click here.

Problem Set 1. First-Order Differential Equations: Separable, Linear

Problem Set 2. Homogeneous Differential Equations

Problem Set 3. Bernoulli's Differential Equations

Problem Set 4. Exact Differential Equations 1

Problem Set 5. Exact Differential Equations 2

Problem Set 6. Harmonic Motion

Problem Set 7. Second-Order Linear Differential Equations and Linear Algebra

Problem Set 8. Non-Homogeneous Second-Order Differential Equations: The Method of Undetermined Coefficients

Problem Set 9. Non-Homogeneous Second-Order Differential Equations: The Method of Variation of Parameters

Problem Set 10. The Laplace Transform: Introduction

Problem Set 11. The Laplace Transform: Transforms of Derivatives

Problem Set 12. The Laplace Transform: The Inverse Transforms

Problem Set 13. The Laplace Transform: Solving Differential Equations

Problem Set 14. The Laplace Transform: Convolution

Problem Set 15. The Laplace Transform: DIfferential Equations with Variable Coefficients

George B. Arfken, Hans J. Weber, Frank Harris, Mathematical Methods for Physicists, 6th Edition, Academic Press

David Betounes, Partial Differential Equations for Computational Science with Maple and Vector Analysis, TELOS, Springer-Verlag New York, Inc.

William E. Boyce, Richard C. Diprima, B. Meade, Elementary Differential Equations, 11th Edition, Wiley

Dean G. Duffy, Green's Functions with Applications, Chapman & Hall/CRC, 2001

Stanley J. Farlow, Partial Differential Equations for Scientists and Engineers, Dover Publications

Erwin Kreyszig, Advanced Engineering Mathematics, Wiley

Peter V. O'Neil, Advanced Engineering Mathematics, WadsWorth