Differential Equations

  1. Higher-Order Linear Ordinary Differential Equations
    1. Preliminary Theory
      1. Initial Value Problems for Higher-Order Equations
        1. Boundary Value Problems for Higher-Order Equations
          1. Homogeneous Linear Equations
            1. Principle of Superposition
              1. Linear Independence of Solutions
                1. Linear Dependence of Solutions
                  1. The Wronskian Determinant
                    1. Definition and Calculation
                      1. Test for Linear Independence
                        1. Properties of the Wronskian
                      2. Nonhomogeneous Linear Equations
                        1. Structure of General Solution
                          1. Complementary Solution
                            1. Particular Solution
                          2. Reduction of Order Method
                            1. Theoretical Foundation
                              1. Application with One Known Solution
                                1. Step-by-Step Procedure
                                  1. Examples and Applications
                                  2. Homogeneous Linear Equations with Constant Coefficients
                                    1. Characteristic Equation Method
                                      1. Formation of Characteristic Equation
                                        1. Auxiliary Equation
                                        2. Case Analysis by Root Types
                                          1. Distinct Real Roots
                                            1. Solution Form
                                              1. General Solution Construction
                                              2. Repeated Real Roots
                                                1. Solution Form with Polynomial Terms
                                                  1. Multiplicity Considerations
                                                  2. Complex Conjugate Roots
                                                    1. Euler's Formula Application
                                                      1. Real-Valued Solution Forms
                                                    2. Higher-Order Equations
                                                      1. Extension to nth-Order
                                                        1. Multiple Root Cases
                                                      2. Nonhomogeneous Linear Equations
                                                        1. Method of Undetermined Coefficients
                                                          1. Applicable Right-Hand Side Functions
                                                            1. Trial Solution Selection
                                                              1. Superposition Approach
                                                                1. Modification for Repeated Roots
                                                                  1. Annihilator Approach
                                                                  2. Method of Variation of Parameters
                                                                    1. General Theory
                                                                      1. Wronskian in Solution Formula
                                                                        1. Application to Second-Order Equations
                                                                          1. Extension to Higher-Order Equations
                                                                        2. Applications of Second-Order Linear Equations
                                                                          1. Simple Harmonic Motion
                                                                            1. Free Undamped Oscillations
                                                                              1. Amplitude and Phase
                                                                                1. Frequency and Period
                                                                                2. Damped Oscillations
                                                                                  1. Damping Coefficient Effects
                                                                                    1. Overdamped Motion
                                                                                      1. Underdamped Motion
                                                                                        1. Critically Damped Motion
                                                                                          1. Spring-Mass-Damper Systems
                                                                                          2. Forced Oscillations
                                                                                            1. External Driving Forces
                                                                                              1. Steady-State Solutions
                                                                                                1. Transient Solutions
                                                                                                  1. Resonance Phenomena
                                                                                                    1. Beating Phenomena
                                                                                                    2. Electrical Circuit Applications
                                                                                                      1. LRC Series Circuits
                                                                                                        1. Mechanical-Electrical Analogies
                                                                                                          1. Current and Charge Relationships
                                                                                                        2. Higher-Order Equations
                                                                                                          1. General Solution Structure for nth-Order
                                                                                                            1. Characteristic Equation for nth-Order
                                                                                                              1. Solution Techniques Extension
                                                                                                                1. Applications to Physical Systems

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                                                                                                              2. First-Order Ordinary Differential Equations

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                                                                                                              4. The Laplace Transform