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 Schrödinger 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