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I'm looking for problems/lessons plans that could be used in a lower-division differential equations course that involve discerning properties of solutions of an equation, IVP, or BVP, without looking for an explicit/implicit solution (general or particular, given the context). Flow lines are an example of this, but I'm looking for something more advanced. One idea I've used is using first-order autonomous equations to figure out the dynamical properties of solutions for different initial conditions. I'm looking for similar ideas. Also: recommendations on how to present existence/uniqueness issues, besides showing a lot of examples, would be appreciated. (Boyce and DiPrima try to give a sketch of a proof of the basic existence/uniqueness result for first order IVPs. I wonder if this can be done without a course on analysis under your belt.)

Bonus: What about introducing group theoretic concepts at an early level? There are textbooks that claim to do this, but I wonder if this is as untenable as trying to teach measure theory in a calculus course.

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    $\begingroup$ What do you think of Arnold's "Ordinary differential equations" for example? $\endgroup$ Jul 4 '11 at 1:25
  • $\begingroup$ I'm actually not familiar with it. Is it suitable for undergraduates? I'm interested in students early exposure to differential equations. I can see the argument for focusing on finding explicit solutions, but it seems to make the subject seem tedious. $\endgroup$ Jul 4 '11 at 1:36
  • $\begingroup$ I am just going to leave a trace from an answer I just deleted which was flagged as being spam, in case it is of use to anyone. It's a project called SIMIODE. There is a link to YouTube videos: youtube.com/channel/UC4y1q6ShHIjRUm7NrM9_dOg $\endgroup$
    – Todd Trimble
    Dec 31 '13 at 15:57
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One thing you can try (and if you decide to do it, I'd like to hear how it goes), is to discuss the differential equation $\frac{dz}{dt} = \alpha z$ with $\alpha \in {\mathbb C}$. The initial condition $z(0) = 1$, of course, corresponds to $e^{\alpha t}$, but what's nice is that you can draw the flow lines and see without calculation what they look like and how they depend on the signs of the real and imaginary parts of $\alpha$. Of particular interest is the case of $e^{it}$ and to explain why it is obvious that $e^{i t} = \cos t + i \sin t$ and $e^{i(t_1 + t_2)} = e^{i t_1} e^{i t_2}$. There's a nice construction of $\pi$ using the symmetries of the vector field -- it suffices to reach $y - x = 0$, and you can prove from the equation satisfied by $y - x$ that this line is reached within time $1$. The uniqueness of solutions implies the rest of the solution can be obtained by reflections. (Even though this equation is "solved explicitly" it can be usefully regarded as the definition of $e^{it}$ and the trig functions.) I suggest it because I think the equation is fundamental, and also it gives the opportunity to visualize where the product $\alpha z$ is compared to $z$.

Some typical exercises which don't (shouldn't) involve solving explicitly are asking asymptotic questions about equations of the form $\frac{dy}{dx} = f(y)$ where $f : {\mathbb R} \to {\mathbb R}$ is some explicit function like a polynomial with some sign changes.

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  • $\begingroup$ Autonomous equations seem like a good source for this type of problem. You could ask to show that a solution of $u''=|u|$ will be non-negative if the initial position and velocity are non-negative. $\endgroup$ Jul 4 '11 at 2:38
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    $\begingroup$ I agree with Mariano's comment on the question that it is strange to accept an answer so fast. I would suggest withdrawing the answer so that other people can share their ideas because I think there should be plenty more. $\endgroup$
    – Phil Isett
    Jul 4 '11 at 3:14
  • $\begingroup$ How do you draw flow lines for complex $\alpha$? It seems like that'd be a 4D picture. $\endgroup$
    – LSpice
    Jul 3 '20 at 18:42
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I have been teaching ODEs for almost 20 years and one thing students not only like to hear, but learn a lot through it, is historical examples of differential equations (i.e., catenary, isochrone, brachystochrone, etc). Good reference for such are the books of G.F Simmons and Hairer & Wanner.

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I am not sure this answer would be helpful since the question is two years old. Anyway, since the question has popped up again, I am happy to introduce this lovely short book that hopefully will give you some ideas: Ordinary Differential Equations: A Brief Eclectic Tour, written by David A. Sánchez.

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