Rigorous Differential Equations text

[SrVishi]

Hello, I am a math major and I was wondering if you guys knew what would be a good rigorous differential equations text. I really like rigor (like Rudin analysis style rigor or whatnot), instead of the typical books that just focus on the method. I want the proofs and everything. I also would like theoretical questions if possible too, and applied as well (I guess it would be hard to find a differential equations text without applications haha). Could anyone tell me if there are any such books? If not, could you at least tell me what your favorite is? I hear Differential Equations with historical notes by Simmons is a good one. Thanks in advance for any response. Sorry if this question is worded weirdly. I'm on mobile in public and can't focus well.

 

[micromass]

What do you know of differential equations already?

 

[SrVishi]

I already had an introductory course in Ordinary Differential Equations, though it was kind of a crash course, so I would kind of like to review it, with a more rigorous flavor of course.

 

[micromass]

Well, you probably can't do better than Ross's differential equations: https://www.amazon.com/dp/0471032948/?tag=pfamazon01-20 This is one of my favorite books out there. Be sure to get the 800 page version, since it contains more theoretical stuff.

Now, the book is very good, and contains quite a lot of theory, with of course the existence and uniqueness theorems. But it isn't really comparable to Rudin in terms of sophistication. But I feel that one should understand the material in this book before getting to a more advanced work, since that advanced work will likely take a lot of things for granted that are in such "introductory books".

For a more rigorous and advanced book, I recommend Teschl: https://www.mat.univie.ac.at/~gerald/ftp/book-ode/ which is freely available on his website, but you can buy it too: https://www.amazon.com/dp/0821883283/?tag=pfamazon01-20 This book will not care about solving differential equations though.

 

[Xiuh]

Another one worth considering is Arnol'd
https://www.amazon.com/dp/0262510189/?tag=pfamazon01-20
In this book you'll get to see some nice connections between differential equations and geometry.

 

[MidgetDwarf]

You can start with Coddington ( introduction), I am familiar with this one. Then you can try his differential equation book(have not looked at it yet, my understanding is not there yet).