# Differential Equations and Their Applications

##### Chapter 1: First-Order Differential Equations
1.2 First-Order Linear Differential Equations

1.3 The Van Meegeren Art Forgeries
1.4 Separable Equations
1.5 Population Models
1.6 The Spread of Technological Innovations
1.7 An Atomic Waste Disposal Problem
1.8 The Dynamics of Tumor Growth, Mixing Problems, and Orthogonal Trajectories
1.9 Exact Equations, and Why We Cannot Solve Very Many Differential Equations
1.10 The Existence-Uniqueness Theorem; Picard Iteration
1.11 Finding Roots of Equations by Iteration
1.11.1 Newton’s Method
1.12 Difference Equations, and How to Compute The Interest Due on Your Student Loans
1.13 Numerical Approximations; Euler’s Method
1.13.1 Error Analysis for Euler’s Method
1.14 The Three Term Taylor Series Method
1.15 An Improved Euler Method
1.16 The Runge – Kutta Method
1.17 What To Do In Practice
##### Chapter 2: Second-Order Differential Equations
2.1 Algebraic Properties of Solutions
2.2 Linear Equations with Constant Coefficients
2.2.1 Complex Roots
2.2.2 Equal Roots; Reduction of Order
2.3 The Nonhomogeneous Equation
2.4 The Method of Variation of Parameters
2.5 The Method of Judicial Guessing
2.6 Mechanical Vibrations
2.6.1 The Tacoma Bridge Disaster
2.6.2 Electrical Networks
2.7 A Model for the Detection of Diabetes
2.8 Series Solutions
2.8.1 Singular Points, Euler Equations
2.8.2 Regular Singular Points, the Method of Frobenius
2.8.3 Equal Roots, and Roots Differing by an Integer
2.9 The Method of Laplace Transforms
2.10 Some Useful Properties of Laplace Transforms
2.11 Differential Equations with Discontinuous Right-Hand Sides
2.12 The Dirac Delta Function
2.13 The Convolutional Integral
2.14 The Method of Elimination for Systems
2.15 Higher-Order Equations
##### Chapter 3: Systems of Differential Equations
3.1 Algebraic Properties of Solutions of Linear Systems
3.2 Vector Spaces
3.3 Dimension of a Vector Space
3.4 Applications of Linear Algebra to Differential Equations
3.5 The Theory of Determinants
3.6 Solutions of Simultaneous Linear Equations
3.7 Linear Transformations
3.8 The Eigenvalue-Eigenvector Method for Finding Solutions
3.9 Complex Roots
3.10 Equal Roots
3.11 Fundamental Matrix Solutions; $$e^{At}$$
3.12 The Nonhomogeneous Equation; Variation of Parameters
3.13 Solving Systems by Laplace Transforms
##### Chapter 4: Qualitative Theory of Differential Equations
4.1 Introduction
4.2 Stability of Linear Systems
4.3 Stability of Equilibrium Solutions
4.4 The Phase-Plane
4.5.1 L.F. Richardson’s Theory of Conflict
4.5.2 Lanchester’s Combat Models and The Battle of Iwo Jima
4.6 Qualitative Properties of Orbits
4.7 Phase Portraits of Linear Systems
4.8 Long Time Behavior of Solutions; The Poincaré-Bendixson Theorem
4.9 Introduction to Bifurcation Theory
4.10 Predator-Prey Problems; or Why The Percentage of Sharks Caught in The Mediterranean Sea Rose Dramatically During World War I
4.11 The Principle of Competitive Exclusion in Population Biology
4.12 The Threshold Theorem of Epidemiology
4.13 A Model for The Spread of Gonorrhea
##### Chapter 5: Separation of Variables and Fourier Series
5.1 Two-Point Boundary-Value Problems
5.3 The Heat Equation; Separation of Variables
5.4 Fourier Series
5.5 Even and Odd Functions