The course follows the book Differential Equations with Boundary-Value Problems by D. G. Zill, and W. S. Wright, 8th Ed, Cengage Learning, 2012

The other usefull sources, which were helpful to prepare my notes:

By Miroslav Stibor, Zaman University. You can get all the below chapters in one PDF (5 MB):

List of chapters

First order DE

  1. Introduction to differential equations
  2. Solution by separating variables
  3. Solution of linear DE
  4. Solution of exact (total) DE
  5. Solution by substitution
    1. Homogeneous DE
    2. Bernoulli DE
  6. Numerical method to solve first order DE (Euler's method)

Modeling with first order DE

  1. Real-world problems modeling with first order DE

Higher order DE

  1. Introduction to higher order linear DE
  2. Homogeneous linear DE
    1. Reduction of order method (for higher order linear DE with non-constant coefficients)
    2. Homogeneous linear DE with constant coefficients
  3. Nonhomogeneous linear DE with constant coefficients: Undetermined coefficients to find $y_p()$
    1. Undetermined coefficients—Superposition approach
    2. Undetermined coefficients—Annihilator approach
  4. Variation of parameters method to find $y_p()$ from $y_c()$
  5. Cauchy-Euler equation (a special type of linear DE with non-constant coefficients)
  6. System of linear DE with constant coefficients (by means of operator $D$)
  7. Nonlinear DE of higher order (substitution $u=y'$, Taylor series)

Modeling with higher order DE

  1. Linear models, initial value problems
  2. Linear models, boundary value problems
  3. Nonlinear models

Series solution of DE

  1. Power series
  2. Power series at singular points

The Laplace Transform

  1. Definition of the Laplace Transform
  2. Solving DE with the Laplace Transform
  3. Additional properties and operations
  4. Solving partial DE using Laplace transform

Fourier series

  1. Introduction to Fourier series
  2. Solving DE by Fourier series
  3. Fourier transform

Partial differential equations, boundary value problems

  1. Problems from physics
  2. Separable partial differential equation