Instead of directly answering the question of "Do engineers use differential equations?" I would like to take you through some background first and then see whether differential equations are used by engineers.
Years ago when I was working as a design engineer for a shock absorber manufacturing company, my concern was how a hydraulic shock absorber dissipates shocks and vibrational energy exerted form road fluctuations to the car suspension system, how shock absorbers behave in the suspension system, and what are the effects of other elements of this system like the mass of the components, the spring rate of the suspension springs and tires, etc.
My research led me to a paper that used a very complicated nonlinear differential equation to study the functional behavior of hydraulic shock absorbers. Here I have to mention that in those days library and paper research was not as easy as today. Computers and the Internet were not as popular as today, and we had to search through engineering indexes which were annually published books containing abstracts of related published papers and articles and order the ones we wanted, if we could find any.
I am not going to discuss the nonlinear differential equation of hydraulics shock absorber here because it takes its own space and time and, more or less is beyond the scope of this article. All I can say is after reading that paper, I had a vivid idea about the function and behavior of shock absorbers in a suspension system.
Later I used the Gauss-Jordan numerical method to solve differential equation of a damper-spring-mass system (figure 1) in a BASIC program. This way I had a simple simulation program by which I could not only understand the effects of different parameters of the system, but also feel the effects of changing, for instance, spring rate or damping coefficient. This simulating program enabled me to have direct practice of spring mass system as if I were working in a lab with real items. I cannot represent the actual results here because I have retired my old Amiga 500 computer which I used for that program with an up-to-date PC but, the results I got looked like figure 2.