In grad school, I worked with two types of math majors who had decided to work in the engineering department. The first were statistics folks who worked in risk/reliability for the nuclear engineering department. However, most of the math majors I've worked with were developing numerical models. For example, let's say you want to model a plasma. In this case you would solve the Poisson equation for electric field in space, the electron Boltzmann equation, and numerous other equations (including relevant chemistry). This is typically all numerical but there could be transforms implemented depending on the geometry. Check out a CFD book for typical applied math in engineering. I can't speak to much outside of the numerical techniques. I know there are some theoretical types that work on crystal structures for thermoelectric materials, for example. The math required for most (BS level) engineering roles is typically just circuit analysis and controlls/vibrations, which are both just about solving partial differential equations.
I remember doing vibration analysis on plates and other solid bodies utilizing 3-dimensional wave equations (think the "del" operator expanding into three partial differential equations of spatial axes). Somewhere along the way we were using Bessel functions for something...must have been really hairy because I don't remember that much about it. The professor's research was about how plates & shells responded to point impulses. In common language, it was about how submarine hull plates responded to sonar pings. I wonder what THAT was all about?