Differential Equations

A differential equation is a mathematical equation that relates an unknown function with one or more of its derivatives, effectively describing how a quantity changes. As the language of change, these equations are fundamental tools in science, engineering, and economics for modeling dynamic systems, from the motion of planets and the flow of heat to population growth and financial markets. Solving a differential equation means finding the specific function that satisfies the relationship, thereby revealing the underlying behavior of the system being studied.

1. Introduction to Differential Equations
   1. Defining a Differential Equation
      1. Mathematical Formulation
      2. Dependent Variables
      3. Independent Variables
      4. Relationship Between Variables and Their Derivatives
   2. Key Terminology
      1. Order of an Equation
      2. Degree of an Equation
      3. Linearity and Non-linearity
         1. Linear Differential Equations
         2. Nonlinear Differential Equations
         3. Coefficients in Linear Equations
      4. Homogeneous Equations
      5. Nonhomogeneous Equations
      6. Autonomous Equations
      7. Non-autonomous Equations
   3. Classification of Differential Equations
      1. Ordinary Differential Equations (ODEs)
         1. Definition and Characteristics
         2. Examples of ODEs
         3. Order Classification in ODEs
         4. Degree Classification in ODEs
      2. Partial Differential Equations (PDEs)
         1. Definition and Characteristics
         2. Examples of PDEs
         3. Order Classification in PDEs
         4. Mixed Partial Derivatives
   4. Solutions to Differential Equations
      1. General Solution
         1. Definition and Properties
         2. Arbitrary Constants
      2. Particular Solution
         1. Definition and Properties
         2. Specific Initial Conditions
      3. Singular Solution
         1. Definition and Identification
         2. Relationship to General Solution
      4. Family of Solutions
         1. Solution Curves
         2. Parameter Variation
      5. Implicit Solutions
      6. Explicit Solutions
      7. Verifying Solutions
         1. Substitution Method
         2. Checking Boundary Conditions
   5. Initial Value Problems (IVPs)
      1. Definition and Formulation
      2. Initial Conditions
      3. Existence of Solutions
      4. Uniqueness of Solutions
      5. Well-Posed Problems
   6. Boundary Value Problems (BVPs)
      1. Definition and Formulation
      2. Types of Boundary Conditions
         1. Dirichlet Boundary Conditions
         2. Neumann Boundary Conditions
         3. Robin Boundary Conditions
         4. Mixed Boundary Conditions
      3. Comparison with Initial Value Problems
   7. Mathematical Modeling with Differential Equations
      1. The Modeling Process
         1. Problem Identification
         2. Variable Selection
         3. Parameter Determination
         4. Equation Formulation
         5. Solution Interpretation
         6. Model Validation
      2. Direction Fields
         1. Construction of Direction Fields
         2. Slope Field Visualization
         3. Isoclines
      3. Solution Curves
         1. Integral Curves
         2. Qualitative Behavior Analysis
         3. Equilibrium Solutions