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I will have to teach a first course in differential equations. What should I cover in this module? For example, in most books, have Laplace Transforms which is fine but I would not use LT to solve differential equations.

This differential equations module is for sophomore students who have done a calculus module in their first year which covered 1st and second-order DEs. They have come across separable equations, integrating factor, Newton's Law of cooling and solving second order by characteristic equation and particular integral. Both first and second order initial value problems. In addition they have solved these equations by numerical methods. They do a final year module called partial differential equations. I hope this sets the background of the students I will be teaching.

I want to write a course that motivates students and has an impact. What topics and what is the most motivating way to introduce differential equations? I want a well-structured and practical approach to differential equations. It is for mathematics and physics students.

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    $\begingroup$ Differential equations are a topic that can be approached in a plethora of different ways (and addressing all of them would overwhelm). For example, you can focus on analytical solutions (and the systems that allow for them), numerical solutions (and the systems that require them); you can focus on dynamical systems or ways leading to PDEs. All approaches have their merit for the right target audience (and “mathematics and physics” isn’t specific enough). Without further specification in this respect, I am afraid this question will turn into a popularity contest. $\endgroup$
    – Wrzlprmft
    Commented Mar 13, 2023 at 16:54
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    $\begingroup$ As you said you “have to teach” the course, I would suspect that whoever wants you to teach this course also has some preferences on what aspects of the topic is relevant. Or to put it differently: I doubt that you can teach a coherent set of topics that you find most interesting (which I would otherwise recommend). $\endgroup$
    – Wrzlprmft
    Commented Mar 13, 2023 at 16:58
  • $\begingroup$ I agree with @Wrzlprmft: as the question is currently worded, you might get many answers - but to get answers that really fit your course, you'll probably need to provide a lot more context information. $\endgroup$
    – Jochen Glueck
    Commented Mar 13, 2023 at 17:01
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    $\begingroup$ Find what courses your students will take later, and see what those instructors expect the students to know from your course. Even if "I would not use LT to solve differential equations", some of those subsequent courses may, and expect their students to have already seen it. $\endgroup$
    – Gerald Edgar
    Commented Mar 13, 2023 at 20:41
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    $\begingroup$ It's possible some form of Gian-Carlo Rota's "Ten lessons I wish I had learned before I started teaching differential equations" could be the basis for an answer. I'm not sure how to write that answer though. web.williams.edu/Mathematics/lg5/Rota.pdf $\endgroup$
    – Chris Cunningham
    Commented Mar 15, 2023 at 18:05

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