Differential Equations

  1. Systems of Linear First-Order Differential Equations
    1. Introduction to Systems
      1. Motivation and Examples
        1. Vector Notation
          1. System Terminology
            1. Geometric Interpretation
            2. Matrix Formulation
              1. Matrix-Vector Form
                1. Coefficient Matrix
                  1. Homogeneous Systems
                    1. Nonhomogeneous Systems
                      1. Fundamental Matrix
                        1. Wronskian for Systems
                          1. Definition and Calculation
                            1. Linear Independence Test
                          2. Homogeneous Systems with Constant Coefficients
                            1. Eigenvalue-Eigenvector Method
                              1. Eigenvalue Problem Setup
                                1. Characteristic Polynomial
                                2. Solution Cases by Eigenvalue Types
                                  1. Distinct Real Eigenvalues
                                    1. Solution Construction
                                      1. General Solution Form
                                      2. Repeated Real Eigenvalues
                                        1. Geometric vs Algebraic Multiplicity
                                          1. Generalized Eigenvectors
                                            1. Jordan Form Applications
                                            2. Complex Eigenvalues
                                              1. Complex Conjugate Pairs
                                                1. Real-Valued Solutions
                                                  1. Oscillatory Behavior
                                                2. Fundamental Matrix Construction
                                                  1. General Solution Representation
                                                  2. Nonhomogeneous Linear Systems
                                                    1. Structure of General Solution
                                                      1. Method of Undetermined Coefficients
                                                        1. Applicable Forcing Functions
                                                          1. Trial Solution Forms
                                                          2. Variation of Parameters Method
                                                            1. Fundamental Matrix Usage
                                                              1. Particular Solution Formula
                                                              2. Applications and Examples
                                                              3. Matrix Exponential
                                                                1. Definition via Series
                                                                  1. Properties of Matrix Exponential
                                                                    1. Computation Methods
                                                                      1. Diagonalization Method
                                                                        1. Jordan Form Method
                                                                          1. Series Computation
                                                                          2. Solution Representation
                                                                            1. Relationship to Scalar Exponential
                                                                            2. Phase Plane Analysis
                                                                              1. Phase Portrait Construction
                                                                                1. Critical Point Analysis
                                                                                  1. Classification of Equilibrium Points
                                                                                    1. Nodes
                                                                                      1. Stable Nodes
                                                                                        1. Unstable Nodes
                                                                                        2. Saddle Points
                                                                                          1. Spiral Points
                                                                                            1. Stable Spirals
                                                                                              1. Unstable Spirals
                                                                                              2. Centers
                                                                                              3. Stability Analysis
                                                                                                1. Asymptotic Stability
                                                                                                  1. Stability Definitions
                                                                                                    1. Lyapunov Stability
                                                                                                    2. Trajectory Behavior
                                                                                                      1. Applications to Physical Systems

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                                                                                                    5. Series Solutions of Linear Equations

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                                                                                                    7. Numerical Methods for Ordinary Differential Equations