Differential Equations

  1. The Laplace Transform
    1. Definition and Basic Properties
      1. Integral Definition of Laplace Transform
        1. Domain of Definition
          1. Existence Conditions
            1. Piecewise Continuity
              1. Exponential Order
              2. Transform of Elementary Functions
              3. Inverse Laplace Transform
                1. Definition and Uniqueness
                  1. Linearity Property
                    1. Transform Tables Usage
                      1. Partial Fraction Decomposition
                      2. Operational Properties
                        1. Linearity Property
                          1. First Translation Theorem
                            1. s-shifting
                              1. Exponential Multiplication
                              2. Second Translation Theorem
                                1. t-shifting
                                  1. Unit Step Function Applications
                                  2. Scaling Property
                                    1. Transform of Derivatives
                                      1. First Derivative
                                        1. Higher-Order Derivatives
                                          1. Initial Value Incorporation
                                          2. Transform of Integrals
                                            1. Derivatives of Transforms
                                              1. Initial Value Theorem
                                                1. Final Value Theorem
                                                2. Solving Initial Value Problems
                                                  1. Transform Method Steps
                                                    1. Equation Transformation
                                                      1. Algebraic Manipulation
                                                        1. Inverse Transformation
                                                        2. Advantages over Classical Methods
                                                          1. Handling Initial Conditions
                                                            1. Examples and Applications
                                                            2. Transforms of Special Functions
                                                              1. Unit Step Function
                                                                1. Definition and Properties
                                                                  1. Piecewise Function Representation
                                                                  2. Dirac Delta Function
                                                                    1. Definition and Properties
                                                                      1. Impulse Response
                                                                        1. Sifting Property
                                                                        2. Periodic Functions
                                                                          1. Transform Formula for Periodic Functions
                                                                            1. Applications to Periodic Forcing
                                                                          2. Convolution and Applications
                                                                            1. Definition of Convolution
                                                                              1. Convolution Theorem
                                                                                1. Properties of Convolution
                                                                                  1. Applications to Differential Equations
                                                                                    1. Transfer Functions
                                                                                    2. Systems of Linear Differential Equations
                                                                                      1. Matrix Form of Systems
                                                                                        1. Transform Method for Systems
                                                                                          1. Solution of Linear Systems
                                                                                            1. Applications to Coupled Oscillators

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                                                                                          3. Higher-Order Linear Ordinary Differential Equations

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                                                                                          5. Series Solutions of Linear Equations