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I have reasonable undergraduate and graduate background(just ode & pde part is horrible).I want an introductory book that doesn't just list some ways to solve a bunch of equation.I want it to give me ideas necessary to make the ways feel reasonable and also keep things rigorous.I have no problem using different books for rigor and ideas.Also,a book of geometric flavor or connecting geometry to it would be very good.As, geometry is the main reason I am studying it.Thank You.

edit:it's not duplicate.I think I asked for something more than the answer's there provided.

Rodrigo de Azevedo
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Sagnik Biswas
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  • "Differential Equations And The Calculus Of Variations" (L. Elsgolts) is a highly regarded book for beginners by a Soviet author and is freely available at archive: https://archive.org/details/ElsgoltsDifferentialEquationsAndTheCalculusOfVariations – Yuriy S Jul 01 '19 at 08:25
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    I can only speak about ODEs: I really enjoyed Hirsch and Smale's "Differential Equations, Dynamical Systems, and Linear Algebra" (the original 1st edition, NOT the latest one with Devaney), and also Lawrence Perko's "Differential Equations, and Dynamical Systems". Both these books complement each other very well, because there is quite some overlap amongst the two; so if one book confuses you, you can refer to the other. Both of these books emphasise linear algerba heavily, and they focus a lot on solving linear systems, interpreting the solutions geometrically, drawing phase portraits etc – peek-a-boo Jul 01 '19 at 15:09
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    They don't just hand you a bag of tricks, and say "use this in so and so circumstance". They really try to present the subject as a collection of interrelated ideas, progressing from the linear case to non-linear case. I think there is a nice balance between the "definition theorem proof" style, and the preliminary motivation for what's going on. – peek-a-boo Jul 01 '19 at 15:18
  • @peek-a-boo,Thank you,this is the kind of book I was looking for. – Sagnik Biswas Jul 04 '19 at 03:59
  • @SagnikBiswas Another thing I should mention is that these books are meant as a first introduction to the subject, so not everything is stated in full generality. If for some reason you would like a small introduction into the slightly more analytical part of ODEs from a very general perspective, then you should take a look at Henri Cartan's Differential Calculus. The second chapter introduces the basic theory of ODEs (existence, uniqueness etc) in the general setting of Banach Spaces. – peek-a-boo Jul 04 '19 at 04:09
  • @peek-a-boo Would you recommend Hirsch and Smale for Dynamical Systems also? Or,there are better approaches for graduate students? – Sagnik Biswas Jun 22 '20 at 03:16
  • I'm not sure what books would be suitable for graduate students. Also, it depends on what topics specifically you're looking to study. Hirsch and Smale is a very good introductory book, which occasionally introduces material from a more abstract perspective than is usually found in other introductory texts. Anyway, if you're looking for something specific, you're probably better off creating a new question – peek-a-boo Jun 22 '20 at 03:24
  • I don't think this question is a duplicate of the other one. I think it is a real issue that lots and lots of ODE books are a "bag of tricks" devoid of educational (or even mathematical) value (cf. Rota's essay). The way this question is framed is particularly important for other mathematically mature audience who wants an actual reference for ODE (that is not a "bag of tricks"). – cicolus Nov 29 '22 at 21:18

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You can try the following references:

$(1)\quad $ "Differential Equations" by Shepley L. Ross

$(2)\quad $ "Differential Equations with Applications and Historical Notes " by George Simmons

$(3)\quad $ "Differential Equations: Theory, Technique, and Practice" by George F. Simmons and Steven G. Krantz

$(4)\quad $ "Elements of partial differential equations" by Ian Sneddon

Besides for theory with solution (Practices purpose) you can also follow

$(5)\quad $ "Ordinary and Partial Differential Equations", and "Advanced Differential Equations" by M. D. Raisinghania

$(6)\quad $ "Differential Equations" by J. G. Chakravorty and P. R. Ghosh

$(7)\quad $ "An Introduction to Differential Equations" by Ram Krishna Ghosh and Kantish Chandra Maity

$(8)\quad $ "Differential Equations" by Richard Bronson

nmasanta
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For more conceptual/geometric perspectives on PDE's, that deviate somewhat from the common analysis views, there are

Michael Bächtold
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