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Week 9 Long Day: Linear First-Order Differential Equations
One of the most important mathematical tools available to engineers are differential equations and their solutions. Many phenomena from the real world can be described by differential equations. Today we will be looking at first-order linear differential equations. These are known from highschool, but with our new knowledge of linear algebra we can now become able to analyze such equations using linear maps between functional vector spaces. In this way we can get a clear overview of the structure of the solutions and we can introduce new methods for their solutions.
Today’s Key Concepts
Function vector spaces. Linear differential equations. The solution formula. Initial conditions. Structural theorem. “The guessing method”. Superposition. Existence and uniqueness of solutions.
Preparation and Syllabus
Today’s topics cover eNote 16 First-Order Linear Differential Equations.
Maple Ressources
dsolve
can solve differential equations with and without initial conditions.
Today’s Maple demo is FirstOrderDiffEquations. Also, in exercise 6 today you will be working with a realistic modelling problem using an interactive eMaple document.
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
Mid-term evaluation
Today the teachers will ask for feedback to them and to the weekly group work with exercises. This evaluation will take place in your usual study areas from about 13:15.
Group Exercises
- The General Solution Formula. By Hand
- The Structural Theorem
- The Solution Formula or the Structural Theorem
- Superposition
- Solution and Visualization using Maple
- Modelling of a Physical Problem
- Differential Equations involving Complex-Valued Functions. By Hand
- Linear Mapping between Functional Spaces
- Linear and Nonlinear Differential Equations
Theme Exercise 3
On Friday this week there is no lecture but you should instead work with Theme Exercise 3. 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.