An introductory mathematical treatise of the theory of ODEs
that is well-accessible for physicists is given by
J. David Logan:
A First Course in Differential Equations (Springer, 2015)
PDF from author
Beware that there were complaints about typos in the Amazon Reviews.
The text book that has fundamentally shaped my understanding of Ordinary Differential Equations is
Vladimir I. Arnol’d:
Ordinary Differential Equations (Springer, 1992)
ISBN: 978-3-540-34563-3
The first chapter should be accessible based on a the introductory maths courses. Some of the advanced material is a challenge even for my graduate students!