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Differential Equations: A Modeling Approach
Glenn Ledder, University of Nebraska - Lincoln

Table of Contents

1 INTRODUCTION
1. Differential Equations and Solutions
2. Mathematical Models and Mathematical Modeling
3. Case Study?-A Successful Volleyball Serve

2 BASIC TECHNIQUES AND CONCEPTS
1. Natural Decay
2. Separable First-Order Equations
3. Slope Fields
4. Euler?s Method
5. Error and the Modified Euler Method
6. Existence of Unique Solutions
7. Case Study?-Scientific Detection of Art Forgery

3 HOMOGENEOUS LINEAR EQUATIONS
1. Linear Oscillators
2. Linearity
3. Homogeneous Equations with Constant Coefficients
4. Real Solutions from Complex Characteristic Values
5. Multiple Solutions for Multiple Characteristic Values
6. Other Homogeneous Linear Equations
7. Case Study?-Chemotherapy for Cancer Treatment

4 NONHOMOGENEOUS LINEAR EQUATIONS
1. General Solutions for Nonhomogeneous Equations
2. The Method of Undetermined Coefficients
3. Forced Linear Oscillators
4. Solution of First-Order Linear Equations
5. Particular Solutions by Variation of Parameters
6. Case Study-?A Tuning Circuit for a Radio

5 GRAPHICAL METHODS FOR AUTONOMOUS PROBLEMS
1. Population Models
2. The Phase Line
3. The Phase Plane
4. Equilibrium Points and Stability
5. Qualitative Analysis
6. Case Study?-A Self-Limiting Population

6 SYSTEMS OF LINEAR EQUATIONS
1. A Linear Algebra Primer
2. Straight-Line Solutions
3. Systems with Real Eigenvalues
4. Systems with Complex Eigenvalues
5. Additional Solutions for Multiple Eigenvalues
6. Local Analysis of Critical Points for Nonlinear Systems
7. Case Study?-Dynamics of Contagious Diseases

7 THE LAPLACE TRANSFORM
1. More on Linear Oscillator Models
2. Definition and Properties of the Laplace Transform
3. Solution of Initial Value Problems with the Laplace Transform
4. Differential Equations with Discontinuous or Impulsive Forcing
5. Case Study?-TBA

8 VIBRATING STRINGS AND TRAVELING WAVES
1. Transverse Vibration of a String
2. Vibration Modes of a Finite String
3. Motion of a Plucked String
4. Case Study-?TBA

SOME ADDITIONAL TOPICS
1. Using Integrating Factors to Solve First-Order Linear Equations
2. Runge-Kutta Methods
3. Proof of the Existence and Uniqueness Theorem for First Order Equations
4. Power Series Solutions
5. Nonautonomous Dynamical Systems
6. Fundamental Matrices
7. Nonhomogeneous Linear Systems
8. Regular Perturbation
9. Projectiles in Sports








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