Ordinary differential equations? Okay: Just freshman calculus actually is a good start. E.g., for viral growth consider y'(t) = k y(t) ( b - y(t) ) where t

Ordinary differential equations? Okay: Just freshman calculus actually is a good start. E.g., for viral growth consider y'(t) = k y(t) ( b - y(t) ) where t is time, y(t) the size at time t, and the rate of growth y'(t) = d/dt y(t) is proportional to both the current size y(t) and the size ( b - y(t) ) yet to be achieved. So, e.g., the growth rate is proportional to the number of present customers talking y(t) and the number of potential customers listening ( b - y(t) ). We assume that the present is time t = 0 and we have the eventual size b and the current size y(0). Then, sure, we have an initial value problem (that is, we know y(0)) for a first order, linear ordinary differential equation. But all that is needed for a solution is just freshman calculus. It's just a routine exercise. Trivial? Once that equation kept FedEx from going out of business. Polished, elegant, insightful, balanced, expert, great first text: Earl A. Coddington, 'An Introduction to Ordinary Differential Equations', Prentice-Hall, Englewood Cliffs, NJ, 1961. For when you want to take some really big next steps up past nearly everyone else in mathematics: Earl A. Coddington and Norman Levinson, 'Theory of Ordinary Differential Equations', McGraw-Hill, New York, 1955. For one of the larger reasons to be interested in ordinary differential equations: Michael Athans and Peter L. Falb, 'Optimal Control: An Introduction to the Theory and Its Applications', McGraw-Hill Book Company, New York, 1966. Also E. B. Lee and L. Markus, 'Foundations of Optimal Control Theory', ISBN 0471-52263-5, John Wiley & Sons, New York, 1967. Once in graduate school I got a reading course to give a lecture a week from Coddington and Levinson, Athans and Falb, and Lee and Markus. The prof didn't show up again after the first lecture. At one time our current Fed Chair Janet Yellen indicated that she saw some potential in using control theory to help manage the economy. And a really sweetheart application of modern control theory, doing amazing things with automatic control of quadrocoptors: https://www.youtube.com/watch?v=w2itwFJCgFQ https://www.youtube.com/watch?v=w2itwFJCgFQ A broader view of some of what can be done with ordinary differential equations, and a long, gorgeous desert buffet of applied math: David G. Luenberger, 'Optimization by Vector Space Methods', John Wiley and Sons, Inc., New York, 1969. Generally, though, apparently the big glory days of ordinary differential equations were for the US DoD and NASA during the Cold War and the Space Race. Likely there are plenty of people now, with gray hair, who supply the expertise needed for current aerospace, etc. Numerical solutions? That field is also nicely developed. Partial differential equations? That's related but quite different.