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I'm a undergraduate student of physics and due of the new coronavirus outbreak i'm stuck at home and i like to begin my studies on ODE. And i like to know some bibliography to where a i can start my studies. To reference i studied until now about differentiation (normal and partial) and integration until line integral. I have studied too an introduction to linear algebra.

vidyarthi
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Gabriel Marino
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    Generally there's two flavors of ODE class. One that either assumes familiarity with linear algebra or introduces you to it, and another that requires a real analysis background. I liked my introductory textbook on the subject by Dennis Zill because of its selection of problems, but others may have better suggestions for a more well written text at that level. If you have the time learning linear algebra first would be more useful to you for ODE and in the long run. – Ninad Munshi Apr 07 '20 at 05:42
  • @NinadMunshi oh my bad, i don't knew that is necessary to know linear algebra to studies ODE and i have studied an introduction to linear algebra. – Gabriel Marino Apr 07 '20 at 05:54
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    it's not necessary in the beginning but eventually a lack of linear algebra background will handicap your ability to pursue further ODE/PDE studies. – Ninad Munshi Apr 07 '20 at 05:57
  • Your university will tell you which book is recommended for the course. – sammy gerbil Apr 07 '20 at 06:39
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    To get your feet wet before proceeding to a full-fledged ODE text, try an ODE chapter at the back of a calculus text, which a few texts in the 1960s and 1970s had (and being older, are often freely available on the internet). For example, Chapter XXI in Calculus with Analytic Geometry by Angus E. Taylor and Chapters 75 & 76 in Schaum's Outline ... Differential and Integral Calculus. – Dave L. Renfro Apr 07 '20 at 08:11
  • Boyce and DiPrima is a good, very widely used textbook. I also recommend Differential Equations and Linear Algebra by Gilbert Strang. – littleO Apr 07 '20 at 09:07

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A good beginner's book from Physicists'(or applied) viewpoint is Kreyszig, Advanced Engineering Mathematics. Then, you could supplement it with books of more theoretical falvour, like Boyce-Di-prima, Differential Equations or the master book Norman Levinson's Ordinary Differential Equations

vidyarthi
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Perhaps a little unconventional, I propose to consider Hairer-Nørsett-Wanner: "Solving ODE I: non-stiff problems". While it is a book on numerical methods for ODE, it starts with a large theory part touching all the non-exotic topics of ODE theory, and is then concerned with the production of numerical solutions. This could be a good source to get an intuition for "real life" ODE.

And for the advanced work, Arnol'd "Mathematical methods in classical mechanics" (this is probably not the easiest classic).

See What is Gian-Carlo-Rota really saying about DE courses? (For students like me), How should a DE course be re-written, if Gian-Carlo Rota is correct? on a critique by a lecturer, Gian Carlo Rota, that wrote a book and gave courses on ODE, becoming increasingly doubtful on their results.

Lutz Lehmann
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From the very beginning!

OK, that was partly tongue-in-cheek. For someone of your level (introductory) and context (physics), I think an informal, intuitive approach is best.

To this end I'll suggest using books that don't get too hung up about free use of differentials (most of the older books satisfy this criterion; the newer ones talk about differentials only in a course on manifolds, and that in a very sluggish, clunky way) and that emphasize applications to physics -- after all that's the origin of most problems in differential equations. To enjoy this subject well, you should have more than a passing familiarity with the differential calculus, and also the integral calculus.

A book that digs dip immediately without being opaque is the Shaum's Outlines on Advanced Mathematics for Engineers and Scientists. But again, look for the older books on the internet, and enjoy. One that comes to mind is Euler's Foundations of the Differential Calculus. Don't think most of it is a rehash of calculus -- he dips into differential equations towards the end in a way that I think shows seamlessly how this is just an extension of the investigations of the calculus, albeit having blown into its own separate field today.

Allawonder
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if you don't mind straying away from books, i would recommend you follow MIT's 18.03 course on OCW along with 3blue1brown's series on Differential equations.

Simo
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