Differential equations

McGraw Hill
Pearson
Cengage
Houghton Mifflin Harcourt
John Wiley & Sons
Dummies (Wiley)

Introduction to Differential Equations

1. Terminology
2. Initial-value problems
3. Mathematical models

First-order Differential Equations

1. Solution curves without a solution
• Direction fields
• Autonomous first-order ODEs
2. Seperable variables
3. Linear equations
4. Exact equations
5. Solutions by substitutions
6. Numerical method

Modeling with First-order Differential Equations

1. Linear models
2. Nonlinear models
3. Modeling with systems of first-order ODEs

Higher-order Differential Equations

1. Linear equations
• Initial-value and boundary-value problems
• Homogeneous equations
• Nonhomogeneous equations
2. Reduction of order
3. Homogeneous linear equations with constant coefficients
4. Undertermined coefficients - superposition
5. Undetermined coefficients - annihilator
6. Variation of parameters
7. Cauchy-euler equation
8. Solving systems of linear DEs by elimination
9. Nonlinear differential equations

Modeling with Higher-order Differential Equations

1. Linear models: initial-value problems
• Spring system - free undamped motion
• Spring system - free damped motion
• Spring system - driven motion
• Series circuit analogy
2. Linear models: boundary-value problem
3. Nonlinear models

Series Solutions of Linear Equations

1. Solutions about ordinary points
• Review of power series
• Power series solutions
2. Solutions about angular points
3. Special functions
• Bessel's equation
• Legendre's equation

Laplace Transform

1. Definition of the Laplace Transform
2. Inverse transforms and transforms of derivatives
• Inverse transforms
• Transforms of derivatives
3. Operations I
• Translation on the s-axis
• Translation on the t-axis
4. Operations II
• Derivatives of a transform
• Transforms of integrals
• Transform of a periodic function
5. Dirac delta function
6. Systems of linear differential equations

Systems of Linear First-order Differential Equations

1. Theory - linear systems
2. Homogeneous linear systems
• Distinct real eigenvalues
• Repeated eigenvalues
• Complex eigenvalues
3. Nonhomogeneous linear systems
• Undertermined coefficients
• Variation of parameters
4. Matrix exponential

Numerical Solutions of Ordinary Differential Equations

1. Euler methods and error analysis
2. Runge-kutta method
3. Multistep methods
4. Higher-order equations and systems
5. Second-order boundary-value problems