This question has been posted before, but I need book with specific qualifications. I do not need books for engineers, book that is centered around calculations and stuff. I need to find a book that is theoretical, proves the statements and has good presentation of the theoretical structure. I have had the book by Tenenbaum, I did not like it. I would be very very thankful if someone shared their knowledge with us about this matter.
The best such book is Differential Equations, Dynamical Systems, and Linear Algebra.
You should get the first edition. In the second and third editions one author was added and the book was ruined.
This book suppose very little, but 100% rigorous, covering all the excruciating details, which are missed in most other books (pick Arnold's ODE to see what I mean). It covers a lot, with the emphasizes on the theory and not on the computational side. However, a lot of exercises will teach you also to compute.
I cannot find enough praise for this book.
You could always try Ordinary Differential Equations by Tenenbaum. It's very comprehensive and thorough.