Reddit reviews Differential Equations, Dynamical Systems, and an Introduction to Chaos

If you want to learn a modern (i.e., dynamical systems) approach, try Hirsch, Smale and Devaney for an intro-level book and Guckenheimer and Holmes for more advanced topics.

> a more Bourbaki-like approach

Unless you already have a lot of exposure to working with specific problems and examples in ODEs, it's much better to start with a well-motivated book with a lot of interesting examples instead of a dry, proof-theorem style book. I know it's tempting as a budding mathematician to have the "we are doing mathematics here after all" attitude and scoff at less-than-rigorous approaches, but you're really not doing yourself any favors. In light of that, I highly recommend starting with Strogatz which is my favorite math book of all time, and I'm not alone in that sentiment.

Ordinary Differential Equations and Dynamical Systems by Gerald Teschl is a really good intro to ODE theory on the first-year graduate level. It also has the benefit of being freely available online. At the undergrad level, I haven't used this book personally but Differential Equations, Dynamical Systems, & and Introduction to Chaos by Hirsch, Smale, and Devaney seems to be a common choice.

For PDE, there are lots of standard texts that don't take the "toolbox" approach: at the undergrad level you have Walter Strauss, and at the begininning graduate level you've got Evans and Folland. For a slightly more advanced treatment, I like John Hunter's PDE notes, also free online.

Prerequisites: you should have a firm grasp of introductory analysis, say at the level of Baby Rudin, before diving into either of these subjects. You should also know your undergraduate linear algebra well.

I would check out Differential Equations, Dynamical Systems, and Linear Algebra by Hirsch and Smale (note this is different from Differential Equations, Dynamical Systems, and an Introduction To Chaos by Hirsch, Smale, and Devaney, which is a less self-contained, less rigourous, and more application-driven sequel).

The former book does rigourous proofs of all the results. It does applications as well, and is actually good at explaining things intuitively as well as rigourously. If you're okay with multivariable calculus, then I think you'd be okay with this book. While it's definitely easier if you already know linear algebra and analysis, this book doesn't assume those as prerequisites (the necessary linear algebra is mostly contained in the book, but the analysis results are usually stated without proof before being used to prove something else). That said, generally I would recommend that one knows linear algebra and real analysis, in addition to multivariable calculus, before reading this book or any other serious book on dynamical systems. They say in the introduction that a strong sophomore could handle this book, but that it's written more for an upper level undergrad or even graduate course.

This is my favourite linear algebra book. This covers all of Calculus, Linear Algebra, and introduces you to ODEs.

Now that might sound like a little bit much, but when learning Linear Algebra you should learn it at the same time you are taking 3 dimensional calculus(Calc 3).

Smale, Hirsh