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Week 12, Long Day: Systems of Differential Equations

In a previous week we worked with first-order linear differential equations. It is very common to have not just one but several of such differential equations. If they are coupled then they can’t be solved individually, and instead we are now dealing with a system of differential equations.

Today we will be working with such systems of first-order linear differential equations with constant coefficients. We call them coupled when unknown functions appear in several of them and we must then develop methods to solve them simultaneously, meaning methods to solve a system of them as a whole. The methodology becomes possible with our work with eigenvalues and diagonalization.

Today’s Key Concepts

Systems of first-order linear differential equations with constant coefficients. System matrix. The method of diagonalization. Homogeneous and inhomogeneous systems. Linear solution structure. Existence and uniqueness of solutions.

Preparation and Syllabus

Today’s topics cover eNote 17 Systems of Linear First-Order Differential Equations.

Maple Ressources

  • The already known dsolve command is useful not just for single differential equations but also for systems of them.

Today’s Maple demo is DiffEquationSystems. Also exercise 6 today will work with an interactive eMaple document with a real-life modelling problem.

Activity Program

  • 10.00 – 12.00: Lecture (aud. 42, b. 303A) (link to streaming)
  • 12.30 – 17.00: Group exercises in the study areas (b. 302, bottom floor)
  • 13.00 – 16.00: Your teachers are present in the study areas
  • 15.30 – 16.00: The teachers will answer conceptual questions to the homework

Group Exercises

  1. Homogeneous Systems of Linear Differential Equations
  2. Homogeneous System of Linear Differential Equations
  3. Structural Theorem. Theory
  4. Structural Theorem. By Hand
  5. Structural Theorem. Maple
  6. Modelling of a Physical Scenario
  7. Existence and Uniqueness. Advanced

Theme Exercise 4

On Friday this week there is no lecture but you should instead work with Theme Exercise 4. The material for the Theme Exercise will be uploaded to Learn at 17:00 today. The Theme test in Möbius opens for answers on Friday between 14:00-17:00. Your Theme result will be included in your Homework portfolio grade. Make sure that you are in a group of 4–6 students and begin working on the Theme exercise early.