Henri Cartan's course in differential calculus does cover quite a few useful things for differential equations, from a high-level point of view : you'll find the notion of differentiation in a generic form, the big theorems are proven (local inversion, Cauchy-Lipschitz, ...).

For the low-level and the explicit, Arnold's "Ordinary differential equations" is a must-read, as Qiaochu Yuan already pointed out.