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I generally use Rudin's book to prepare for my analysis lectures, however, we started doing Lagrange multipliers and differential equations (e.g. Picard-Lindelöf Theorem) which unfortunately isn't covered in Rudin.

I am now looking for a book (perhaps similar to Rudin) which covers these topics. Recommendations would be much appreciated.

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There is a book by Arnold, and another one by Walter, title being roughly ordinary differential equations. (For Arnold don't take the one "geometrical methods in the theory of ODE's", this is much more advanced. just ODE's.) But I don't think that they are similar to Rudin, but the Picard-Lindelöf theorem can be found in almost any book which covers the Banach conraction principle. For less specific books consider e.g. Zorich's analysis.

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