I'd go for Tenenbaum and Pollard's book Ordinary Differential Equations. It has whole chapters devoted to applications. After that, you might consider Farlow's book Partial Differential Equations for Scientists and Engineers. That one also has lots of methods, and includes Fourier Transforms.
Are you also interested in book recommendations for quantum mechanics? I don't know much about electronics or Fourier Transform NMR spectroscopy, so I wouldn't be a good person to recommend in those fields.