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Class Differential Equations

  • Presentation

    Presentation

    In many situations in mathematics and its applications, it is natural to consider models that establish relationships involving not only variables, but also their variations. The equations that contain derivatives of their unknowns are called differential equations. In this curricular unit, a brief introduction to the quantitative theory of ordinary differential equations is made through the learning of tools and methods of solving them. Some of the examples and models presented have a framework in the areas of Engineering, Chemistry, Physics, Biology and Economics.

  • Code

    Code

    ULHT39-8450
  • Syllabus

    Syllabus

    1. Introduction

    - Mathematical models

    - Direction fields

    - Classification of differential equations

    - Solution of a differential equation

    - Initial value problem (IVP)

    - Existence and uniqueness of solution of an IVP

     

    2. First-order differential equations

    - Linear equations

    - Separable equations

    - Modeling real systems

    - Linear versus nonlinear differential equations

    - Autonomous differential equations and populations dynamics

     

    3. Second-order linear differential equations

    - Homogeneous equations with constant coefficients

    - Complex roots of the characteristic equation, repeated roots and reduction of order

    - Nonhomogeneous equations, method of undetermined coefficients and method of variation of parameters

     

    4. Systems of first-order linear equations

    - General theory

    - Homogeneous systems with constant coefficients

    - Complex eigenvalues

    - Fundamental matrices

    - Repeated eigenvalues

    - Nonhomogeneous systems

  • Objectives

    Objectives

    - Classify differential equations and identify their order

    - Discuss the existence and uniqueness of solutions of ordinary differential equations

    - Solve first-order differential equations: linear and separable

    - Solve second-order linear differential equations: with constant coefficients, non-homogeneous and with variable coefficients

    - Solve systems of first-order linear differential equations: homogeneous and non-homogeneous

    - Model real systems with differential equations

  • Teaching methodologies and assessment

    Teaching methodologies and assessment

    Creation and availability of short videos on topics related to the subject taught, as well as historical curiosities, playful problems and applications to the real world.

  • References

    References

    - BOYCE, W., DiPRIMA, R., & MEADE, D. (2017). Elementary Differential Equations and Boundary Value Problems (11th ed.). Wiley.

    - KRANTZ, S. G. (2016). Differential Equations: theory, technique, and practice with boundary value problems. CRC Press.

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