Differential Equations self-study

I think Hirsch and Smale's First edition book is absolutely amazing. It progresses nicely starting with linear systems, and generalizing the treatment to non-linear ODEs. It treats the linear algebra clearly, it's very geometric, and the theorems are stated clearly, and proven very nicely. In this book they're not really concerned with the billions of techniques of solving ODEs, rather they are focused on a few key principles, and they elucidate them very nicely. (as you can tell, I'm very fond of this book)

A second book which treats the material in a similar spirit is Lawrence Perko's Differential Equations and Dynamical Systems. I found Perko's and Hirsch/Smale to be very nice complementary texts.

Finally, if you want a fearlessly general glimpse to the subject of ODEs, Henri Cartan's book Differential Calculus has a small chapter devoted to the main setup of the theory; existence, uniqueness, smooth dependence on initial conditions, linear equations etc in the general context of Banach spaces.

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But really, the subject of ODEs is very big, and different books have different goals. The books I mentioned focus on the geometric aspect and linear algebra (for the first two), but if you/the prof have different intentions, then obviously, you should consider a different source.