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Differential Equations

Differential equations for college and university students. Contains direction fields, separation of variables, linear 1st & 2nd order ODEs, LaPlace transforms, and more.

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Course content

Chapter 1: Differential equations

  • Introduction to differential equations
    1. The notion of differential equations
    2. Notation for ODEs
    3. Order and degree of an ODE
    4. Solution of differential equations
    5. Linear ODEs
  • Direction field
    1. Direction field
    2. Euler’s method
    3. Autonomous ODEs
    4. Existence and uniqueness of solutions of ODEs
    5. Solution strategy on the basis of the slope field
  • Separation of variables
    1. Differentials
    2. Differential forms and separated variables
    3. Solving ODEs by separation of variables
  • Linear first-order differential equations
    1. Uniqueness of solutions of linear first-order ODEs
    2. Linear first-order ODE and integrating factor
    3. Solving linear first-order ODEs
  • Linear second-order differential equations
    1. Uniqueness of solutions of linear 2nd-order ODEs
    2. Homogeneous linear 2nd-order ODEs with constant coefficients
    3. Solving homogeneous linear ODEs with constant coefficients
    4. The Ansatz
  • Solution methods for linear second order ODEs
    1. The Wronskian of two differentiable functions
    2. Variation of constants
    3. From one to two solutions
    4. Solving linear second-order ODEs
  • Systems of differential equations
    1. Systems of coupled linear first-order ODEs
  • End of differential equations
    1. Applications of ODEs

Chapter 2: Differential equations and Laplace transforms

  • Differential equations and Laplace transforms
    1. The Laplace transform
    2. The inverse Laplace transform
    3. Laplace transforms of differential equations
    4. Convolution
    5. Laplace transforms of heaviside functions
    6. Laplace transforms of periodic functions
    7. Riemann-Stieltjes intergration
    8. Laplace transforms of delta functions
    9. Transfer and response functions

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