Chapter 1: Introduction to Differential Equations
1.1: What are Differential Equations + Real Life Applications
1.2: Initial Value Problems
Chapter 2: First Order Differential Equations
2.1: Solution Curves, Direction Fields, Autonomous First Order DE’S
2.2: Separable Variables
2.3: Linear Equations
2.4: Exact Equations
2.5: Solutions by Substitutions
2.6: A Numerical Method
Chapter 3: Modeling with First Order Differential Equations
3.1: Linear Models
3.2: Non-Linear Models
3.3: Modeling with Systems of First Order DE’S
Chapter 4: Higher Order Differential Equations
4.1: Linear Equations – Initial Value and Boundary Value Problems
4.2: Linear Equations – Homogenous and Nonhomogenous Equations
4.3: Reduction of Order
4.4: Homogeneous Linear Equations with Constant Coefficients
4.5: Undetermined Coeffcients: Superposition Approach
4.6: Undetermined Coeffcients: Annihilator Approach
4.7: Variation of Parameters
4.8: Cauchy-Euler Equations
4.9: Solving Systems of Linear DE’s by Elimination
4.10: Nonlinear DE’s
Chapter 5: Modeling with Higher Order Differential Equations
5.1: Linear Models – Initial Value Problems with Spring/Mass Systems – Free Undamped Motion, Free Damped Motion, Driven Motion
5.2: Linear Models – Initial Value Problems with Series Circuit Analogue
5.3: Linear Models – Boundary Value Problems
5.4: Nonlinear Models
Chapter 6: Series Solutions of Linear Equations
6.1: Solutions About Ordinary Points – Power Series Solutions
6.2: Solutions About Singular Points
6.3: Special Functions – Bessel’s Equation and Legendre’s Equation
Chapter 7: Laplace Transform
7.1: What is the Laplace Transform
7.2: Inverse Transforms and Transforms of Derivatives
7.3: Operational Properties – Translation on the s-axis and t-axis
7.4: Operational Properties – Derivatives of a Transform, Transforms of Integrals and Periodic Functions
7.5: Dirac Delta Function
7.6: Systems of Linear DE’s
Chapter 8: Systems of Linear First Order Differential Equations
8.1: Homogeneous Linear Systems – Distinct Real Eigenvalues, Repeated Eigenvalues, Complex Eigenvalues
8.2: Nonhomogenous Linear Systems – Undetermined Coefficients and Variation of Parameters
8.3: Matrix Exponential
Chapter 9: Numerical Solutions of Ordinary Differential Equations
9.1: Euler Methods and Error Analysis
9.2: Runge-Kutta Methods
9.3: Multistep Methods
9.4: Higher Order Equations and Systems
9.5: Second Order Boundary Value Problems
Chapter 10: Plane Autonomous Systems
10.1: Autonomous Systems
10.2: Stability of Linear Systems
10.3: Linearization and Local Stability
10.4: Autonomous Systems as Mathematical Models
Chapter 11: Orthogonal Functions and Fourier Series
11.1: Orthogonal Functions
11.2: Fourier Series
11.3: Fourier Cosine and Sine Series
11.4: Sturm: Liouville Problem
11.5: Fourier-Bessel and Fourier-Legendre Series
Chapter 12: Boundary Value Problems in Rectangular Coordinates
12.1: Separable Partial DE’s
12.2: Classical PDE’s and Boundary Value Problems
12.3: Heat and Wave Equations
12.4: Laplace’s Equation
12.5: Nonhomogeneous Boundary Value Problems
12.6: Orthogonal Series Expansions
12.7: Higher Dimensional Problems
Chapter 13: Boundary Value Problems in Other Coordinate Systems
13.1: Polar Coordinates
13.2: Cylindrical Coordinates
13.3: Spherical Coordinates
Chapter 14: Integral Transforms
14.1: Error Function
14.2: Laplace Transform
14.3: Fourier Integral
14.4: Fourier Transforms
Chapter 15: Numerical Solutions of Partial DE’s
15.1: Laplace’s Equation
15.2: Heat and Wave Equations
Link to Textbook: Differential Equations, 11th Edition by Zill