This [worksheet][1] from Dept. of Mathematics in University of Washington guides through two examples using differential equations: forensic mathematics and spread of rumor.

Also the internet magazine called +plus magazine has [many examples of applications][2] such as mathematics of traffic jam. 


  [1]: http://www.math.washington.edu/~m125/Worksheets/DiffEQ.pdf
  [2]: http://plus.maths.org/content/teacher-package-differential-equations