Dear Mr. Nico Scheerlinck,

I think it might be a good idea to read an interesting article about your topic.

It is the article by Carneiro F.; P. Leão C.; F. C. F. Teixeira S. (2010) Teaching differential equations in different environments: A first approach. Computer Applications in Engineering Education 18(3) pp 555 -562.

I hope I have been of some help.

Best,

Andrija

I would say in addition to suggestions from colleagues above it may be valuable to teach the origin or applications of certain types of differential equations so that the student would have a better appreciation of why they are learning to solve differential equations. For example start off with simple DEs governing chemical kinetics or radioactive decay and proceed to more complicated problems such as the Lotka-Volterra problem, the diffusion problem, etc.

depends on the level and priot knowledge of the students, i would think.

1. starting point: (a) review the nature of derivatives through "zooming-in" on the graph of a function to show "local straightness" for small values of "x". Maybe introduce numerial solutions using a spreadsheet, showing how euler's method can be derived from local straightness; (b) examine examples of how DE arise, including language about "rates of change" and its mathematical representation; (c) review relationship between differential and integration, including general and particular solutions.

2. solving de: use the ideas from 1 to approach solving de by numerical/graphical/analytic means using digital technologies such as Geogebra/Mathematica/Maple/DERIVE. Each digital application has a command for solving ode.Give the students a couple of examples using the command and encourage them to "guess the rule" used by the software. Either by direct instruction or collecting student ideas, link the "computer's" rule to techniques of integration and more formal expressions of solutions.

Clearly this approach can be used to introduce second order des using software and discussion. As a means of consolidating these ideas, get students to work in groups to solve a range of first and second order problems and present their solutions to the other students.

At the core of the approach is the development of students' intutions about change and how it can be modelled for analytic purposes. Many of the problems students experience with odes are because they learn techniques but are not clear where they come from or why they are being used. Digital technologies are useful in helping to visualise and manipulate mathematical objects in this context.

Naturally this approach may be a challenge in the context of a 10 week lecture course, but student evaluations and results suggest it is effective. It was taught in four 2-hour workshops with students working groups on paper-and-pencil and digitally based activities. There were 40 students and 4 tutors available. It was assessed, by both tutors and students, for individual student mathematical product, group interaction and presentations . Final mark was given by the tutors.

Not a answer, but a further question. What would be a first historical example of a differential equation? To be more precise, which is the first, main case of an advance in science such that the idea of a differential equation was needed to make that breakthrough? Are we going back to Newton? (if so, which proposition by Newton?)

My contribution is general as differential equations is not my area.

You need to determine what prior knowledge they have; be it through a pretest or a warm up exercise. Present the intention of the lesson concept and what success is. Sequential worked examples with follow up activities either independant or in small groups. Students to share their methods and solutions where any misconceptions can be overturned by the teacher. Continued practise with the teacher providing feedback and monitoring for understanding and assessment. Post test. Revision to take place over time.

References:

High Impact Teaching Strategies (HITS)

Research and Proven Practices John Hattie

To understand ODEs or PDEs, classes in Engineering and Science are essential. See my Casebook titled "Engineering Design Optimization using Calculus Level Methods- A Casebook Approach", http://fortrancalculus.info/textbooks/ . Their are some 20+ Engineering and Science problems showing their math problems and Optimal solution. Plus a little write-up / intro to each problem.

ODEs may be 'defined' as a picture of a processes interaction. (A picture is worth a thousand words, right? ) For example, what is an apple? You could look up definitions, do google searches, etc. for ever and still not understand what is meant by an Apple. Take a bite of an apple and THEN you have a chance of understanding it... delicious! Take some engineering/science classes and learn those ODEs :)

For ODE solutions, try the FortranCalculus (FC) language that got its start by NASA's Apollo Space program that got us to the moon!

FC-Compiler™ application is a (free) Calculus-level Compiler that simplifies Tweaking parameters in ones math model. FC solves Algebraic through Ordinary Differential Equations Equations; Laplace transforms; etc. FC is based on Automatic Differentiation that simplifies computer code to an absolute minimum; i.e., a mathematical model, constraints, and the objective (function) definition.

Most math Modeling, Simulations, & Optimizations can be easily solved with a Calculus-level language/compiler. A half-hour per problem should be enough time to code and execute most math problems. Solves Algebraic Equations through Ordinary Differential Equations, for more visit http://fortrancalculus.info/apps/fc-compiler.html . Equations may be implicit, non-linear, Boundary Value Problems (BVP), Initial Value Problems (IVP), etc. Download, install, & execute some demo problems. Simple and its Freeware! (PDE solvers are in the works!)