Undergraduate ODE textbook following Rota I imagine many people are familiar with the extremely entertaining article "Ten Lessons I Wish I Had Learned Before I Started Teaching Differential Equations" by Gian-Carlo Rota. (If you're not, do yourself a favor and follow the link I provided.)
I, a number theorist, have been told that I am to teach an undergraduate ODE class one year from now. (Nevermind that my familiarity with ODEs is, to put it mildly, minimal.) In a good faith effort to serve my students as well as possible, I am asking the following question.

Has any ODE textbook been written which addresses and assuages the issues brought forth in Rota's article? 

Is there any way for me to teach ODEs next year and not, having read Rota's article, feel dirty about it?
 A: I like these two differential equations books:


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*Differential Equations and their Applications, by Martin Braun (Amazon)

*Ordinary Differential Equations, by V.I. Arnol'd (Amazon)
I don't claim they meet all Rota's criteria, but I think they are somewhat fun and offer insight.  Braun is especially recommended for examples like how to detect art fraud by the age of the paint, the galloping gertie bridge disaster in tacoma (although he slightly erroneously names the culprit "resonance"), and predator - prey examples involving sharks in wartime.  
The main thing to make ode seem appealing to me is just to point out an ode is a vector field wanting (a family of) parametrizing curves.  Arnol'd offers this sort of geometric appeal.  It was after reading Arnol'd that I understood why Reeb's theorem is so intuitive, (a compact manifold with a function having just two non degenerate singular points is a sphere).
A: My advice would be to teach the traditional course.  Especially because you lack knowledge of ODEs themselves.  Deciding that you are going to do something different when you lack knowledge, experience of the topic and likely sympathy with the objectives of the course (teaching engineers).
Rota has a fun to read essay and there are parts of it that I like.  For example 2nd order ODE with constant coefficients is very common in science and engineering.  however, even here the course really already covers it heavily. In some cases repetitively:  calculus primer on ODEs, ODE itself, and then in majors courses in physics, chemistry, engineering.  I would also emphasize that Rota's book never had great commercial success either as a common text like Brown or an underground Amazon success like Tenenbaum.  Read the reviews of the Rota text on Amazon for some perspective.
Once you are a little more experienced perhaps it makes sense to do something revisionist but I would be very hesitant of the "I'm a number theorist and will teach all the engineers their ODEs differently since it was all wrong before...even though I don't even know much about the topic or about teaching it"
A: When I was a grad student, our department taught ODEs in the "bag of tricks" style.  Guilty!  As a TA I even hammed it up doing an impression of Emeril -- bam! -- saying we were making a recipe book of techniques to attack problems.
But when I was an undergrad and originally learned the material, it was taught in a vastly different way.  I took ODEs with Borrelli at Harvey Mudd.  They referred to their own book often enough; it's more applications oriented.  But we also spent a lot of time with the workbook.
It seems there's a second edition, and Wiley has "companion" sites for the book, instructors, and students.  See:


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*http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471433322.html

*http://bcs.wiley.com/he-bcs/Books?action=index&itemId=0471433322&bcsId=1850

*http://bcs.wiley.com/he-bcs/Books?action=index&itemId=0471433322&bcsId=1849
BTW, the workbook link above is to the CODEE site.  CODEE is the Community of Ordinary Differential Equations Educators.  The CODEE site has a lot of useful resources.
A: I would suggest

ArnoI'd, V.I. (Vladimir lgorevich)
  Geometrical methods in the theory of ordinary
  differential equations.
  Springer (1988)

of course, it's not to be taken "as is" for an undergraduate ODE class, but a lot of modern important examples (taken from nature) are given and discussed. Overall, there are a lot of well illustrating and expressive drawings which can inspire the lectures.   
A: As @AndreiHalanay says, probably the best that one can do here is to give a not-guilty textbook, rather than a perfect one.  I haven't seen Teschl's book that he recommends, but I have read part of the Hirsch–Smale–Devaney one.  Much as I enjoyed it personally, I got the impression it asked a bit too much of students for me to use it.
A book in a similar spirit (no surprise, since it shares an author), but a little less demanding of students—which it accomplishes largely by being less ambitious (those used to teaching out of Boyce and diPrima will be shocked by the fact that, for example, it treats integrating factors only for linear equations)—is Blanchard–Devaney–Hall.  I haven't specifically compared it to Rota's list (which rhymes very well with my woes after having taught the course for years), but I did try going back to Boyce and diPrima once after having taught a few times from BDH, and it really made me appreciate the latter.
EDIT:  Having now refreshed my memory of Rota's essay:  I had forgotten that he specifically disclaims the utility of integrating factors, so I guess my remark about them can be taken as an endorsement of BDH.  I am also disappointed by what seems to me to be BDH's sparse treatment of change of variables; at least they do not explicitly describe the importance of this technique.
A: I am in the same department as Mark Meckes, and have enjoyed using Blanchard, Devaney and Hall over the decade I've been teaching the course.  I appreciate its emphasis on qualitative methods and insights, perhaps because these are the aspects of differential equations I use in my own work (in mathematical neurosocience).  Having just read the Rota essay, I don't see it as giving any very particular recommendations vis-a-vis what should go into a textbook.  The lectures are where I try to bring the material to life for the students, whatever textbook I use, and the course is more than just the lectures and the textbook.  (I recall an evaluation once by an angry student who said if it weren't for doing all the homework problems he would have learned nothing from the course -- as if just listening to lecture and reading the text were supposed to suffice...)
So -- I endorse BDH.
A: There are a number of textbooks which are not as guilty:


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*M.W. Hirsch, S. Smale, R. Devaney, Differential Equations, Dynamical Systems, and an Introduction to Chaos

*G. Teschl, Ordinary Differential Equations and Dynamical Systems
The last one is also freely available here. As the titles say, both take a rather geometric path and give many examples from various sciences. 
The first one waits until the final chapter to give the classical existence and uniqueness theorems.
A: I read Rota's essay today, having already taught the "cookbook" ODE course 5 or 6 times.  While the essay carries a lot of wisdom, I think the answer of Nov 25 '18 at 23:18 by guest is absolutely the most useful.  
With non math majors, you must strike a balance between concrete and conceptual and most of the suggested books are way too advanced for this level (you will lose most of the students).  
Finally, while I guess I subscribe to Rotas arguments, another perspective might be that teaching (some) tricks is not necessarily a bad thing especially if you can deconstruct the tricks for the students.  There is nothing dirty about such an  exercise well done.
A: "ODE tetbook following Rota"  suggests to me
Birkhoff, Garrett; Rota, Gian-Carlo, Ordinary differential equations, Introductions to Higher Mathematics. Boston-New York-Chicago: Ginn and Company, VII, 318 p. (1962). ZBL0102.29901.  
But of course that is too advanced for an undergraduate students (except perhaps at places like MIT).
