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As part of the mathematics program offered at my college, I took an introductory ODE course a few semesters back. This was the one math course in my entire college career that I was totally lost in. Even after pursuing additional information through my instructor and scouring countless other ODE books, I have no idea how the various methods of solutions for ODEs came to be. They are all non-obvious to me--well, all except for separation of variables--and really seem quite contrived.

What is a good book or online resource that conveys ODE solution methods in an easy to understand way? The book I used for my course was "Fundamentals of Ordinary Differential Equations" by Nagle, Saff, and Snider and it was very much written for those people who really only wanted or needed an algorithmic way to find solutions. What I am really looking for are some explanations for why certain methods work, how they work, and how they came to be from a theoretical standpoint, because right now they all seem like lucky guesses and don't seem interconnected at all.

If you can suggest some books or online resources, that would be great. Please provide some rationale for why you think the resource is good. No modeling needs to be covered in the resource as that is rather simple.

user120323
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  • Can't offer you a book; but the most elementary methods are a direct result of basic calculus and linear algebra. So perhaps you can back your way into explanations there. Many of the "contrived" solutions are exactly that. Someone needed to solve the problem and fiddled around with it until they figured something out. If you fiddled around with one of them for 100 or 1000 hours you might find that solution too. – Betty Mock Jan 10 '14 at 05:26
  • I think trying to understand these things better is a good idea. However, ultimately most diff eq's are solved by numerical methods, and the purpose of presenting these elementary approaches is to get you oriented to the field, and give you some intuitive idea of what solutions ought to look like. Finally, you could ask some of your questions here as in "can someone give an intuitive explanation of .. ". People love to answer those. – Betty Mock Jan 10 '14 at 05:28
  • You seem to biased for the intuitive way of doing ODE. I think OP wants to understand how these things derive and arise. Nothing wrong with that. – felasfa Jan 10 '14 at 19:15
  • My impression has always been that many of the methods for solving ODEs involve simply guessing that the solution has a particular form, such as $e^{rx}$, and then just plugging in to figure out what the parameters (like $r$) should be. I think in many cases it really is just a "lucky guess" which we can then prove will always work for a certain class of ODEs. – littleO Jul 09 '15 at 21:47
  • You might be interested in the text by Rota discussed here, also here. For the history of numerical method, a review article by Butcher is discussed here – Lutz Lehmann Aug 25 '20 at 17:14

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I have been in your boat. Trust me you can't go wrong with book

Ordinary Differential Equations, Tenenbaum and Polland. Below is Amazon Link.

http://www.amazon.com/Ordinary-Differential-Equations-Dover-Mathematics/dp/0486649407/ref=sr_1_2?ie=UTF8&qid=1389339215&sr=8-2&keywords=ode

It is a fair balance of theory and applications. It is ideal for self study and as a Dover edition, it is cheap.

If you feel that is not to your level of rigor, consider

Ordinary Differential Equations, Gian Rota

felasfa
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Mostly I prefer ,

  1. Differential equation with application and historical notes by G.F Simmons ( a good book for beginners also it contains a sufficient amount of exercise)
  2. Differential equation by S.L Ross
  3. An introduction to ordinary differential equations by E.A Coddington ( this is for doing Differential equation rigorously )
Siong Thye Goh
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Om Prakash
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Tanenbaum and Pollard is a very good book. I never used it in a course but I bought it for my library. At 800 pages it covers a lot more than you would normally cover in a one semester Diff Eq course.

joseph c koziar
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