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I'm a physics & mathematics (double major) student, and I'm going to do a self-study on differential equations this semester. However, any book that I have looked basically gives the list of some classic type of differential equations, such as ODEs with constant coef., homogeneous ODEs, not homogeneous ODEs etc, and shows how to solve those particular examples with some methods.

However, this does not teach me anything. It just makes me memorise how to solve some particular problems with particular methods, which is something that I really hate to do it.

So my question is that is there any book that you can suggest me such that it treats the subject in a systematic way, and explains the things with reasons and develops the subject in a mathematically correct way instead of just showing how to solve some problems?

Note that, even the graduate level texts are OK as long as it meets the criteria.

I hope I'm able to clearly explain myself. If not, please collaborate.

Edit:

It would be really guiding if you also include your own thoughts.

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I have two suggestions, with the same title (Ordinary differential equations) and from the same publisher (Springer-Verlag): the one by Vladimir Arnold (perhaps best suited for a Physics student) and the one by Wolfgang Walter.

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  • $\begingroup$ +1 for Arnold's although not an easy reading... $\endgroup$ – Miguel Sep 26 '17 at 10:28
  • $\begingroup$ @Miguel I agree. $\endgroup$ – José Carlos Santos Sep 26 '17 at 10:29
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The outstanding Differential Equations, Dynamical Systems and Linear Algebra never goes out of fashion: https://www.amazon.es/Differential-Equations-Dynamical-Systems-Mathematics/dp/0123495504

Of course it deals with linear ODEs, but not giving recipes, but as a model for all ODEs, through linearization and asymptotic analysis.

It looks like there is a new edition and, although I do not know it, I guess it will be likewise extraordinary.

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