Differential Equations

  1. First-Order Ordinary Differential Equations
    1. Separable Equations
      1. Definition and Recognition
        1. Separation of Variables Method
          1. Integration Techniques
            1. Solution Verification
              1. Applications and Examples
              2. Linear First-Order Equations
                1. Standard Form Recognition
                  1. Integrating Factor Method
                    1. Derivation of Integrating Factor
                      1. Calculation Steps
                        1. Solution Process
                        2. Solution Verification
                          1. Applications and Examples
                          2. Exact Equations
                            1. Definition of Exactness
                              1. Test for Exactness
                                1. Partial Derivative Criterion
                                2. Solution Method for Exact Equations
                                  1. Integrating Factors for Non-exact Equations
                                    1. Finding Integrating Factors
                                      1. Special Cases
                                      2. Applications and Examples
                                      3. Substitutions and Transformations
                                        1. Homogeneous Equations
                                          1. Definition and Recognition
                                            1. Substitution Method
                                              1. Solution Process
                                              2. Bernoulli Equations
                                                1. Definition and Recognition
                                                  1. Transformation to Linear Form
                                                    1. Solution Steps
                                                    2. Equations with Linear Coefficients
                                                      1. Recognition Patterns
                                                        1. Appropriate Substitutions
                                                          1. Solution Techniques
                                                        2. Existence and Uniqueness Theory
                                                          1. Picard's Existence and Uniqueness Theorem
                                                            1. Statement of Theorem
                                                              1. Conditions for Application
                                                              2. Lipschitz Condition
                                                                1. Successive Approximations
                                                                  1. Examples of Non-uniqueness
                                                                    1. Counterexamples
                                                                    2. Applications of First-Order ODEs
                                                                      1. Population Dynamics
                                                                        1. Exponential Growth Model
                                                                          1. Exponential Decay Model
                                                                            1. Logistic Growth Model
                                                                              1. Carrying Capacity
                                                                              2. Radioactive Decay
                                                                                1. Half-life Calculations
                                                                                  1. Carbon Dating Applications
                                                                                  2. Newton's Law of Cooling
                                                                                    1. Temperature Modeling
                                                                                      1. Heating and Cooling Problems
                                                                                      2. Mixing Problems
                                                                                        1. Tank Mixing Models
                                                                                          1. Concentration Changes
                                                                                            1. Flow Rate Considerations
                                                                                            2. Simple Electrical Circuits
                                                                                              1. RC Circuit Analysis
                                                                                                1. RL Circuit Analysis
                                                                                                  1. Current and Voltage Relationships

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                                                                                              1. Introduction to Differential Equations

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                                                                                              3. Higher-Order Linear Ordinary Differential Equations