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Braun’s classic text is noteworthy for its combination of rigorous pure mathematical formalism and real-world applications. The mathematical underpinnings of the Tacoma Narrows Bridge collapse, international conflict, syphilis outbreaks, and the propagation of waves are carefully derived and proved. The preponderance formal definitions, lemmas, and theorems do not prevent the interested non-specialist from understanding the models, however, because their qualitative descriptions are clearly presented in plain English prose.
This probably isn’t the best book to self study the math of differential equations—especially if you haven’t completed upper-division real analysis coursework. But Differential Equations and Their Applications is a sterling example of why you would want to complete such a course of study: Braun’s magnum opus leaves little doubt that the Book of Nature is indeed written in the language of diff EQs.