A lot of the equations I use come from the same starting points as you might see in course work.
One neat example is the paraxial Helmholtz equation, which described the evolution/propagation of laser beams.
For example, from Maxwell's equations, we can get the wave equation for light.
\frac{\partial^{2}\vec{E}}{\partial x^{2}}+\frac{\partial^{2}\vec{E}}{\partial y^{2}}+\frac{\partial^{2}\vec{E}}{\partial z^{2}}=\frac{1}{c^{2}}\frac{\partial^{2}\vec{E}}{\partial t^{2}}
If we only consider the amplitude of the electric field E as important, and assume the light is monochromatic enough that the electric field can be factored into a time-dependent function f(t), and a space-dependent function A(x,y,z), the space-dependent part is described by the Helmholtz equation
\frac{\partial^{2}A}{\partial x^{2}}+\frac{\partial^{2}A}{\partial y^{2}}+\frac{\partial^{2}A}{\partial z^{2}}=-k^{2} A
Now, if we also say that the light is predominantly moving along the z direction, or that the z-component of the momentum of the field is much larger than the x or y components, we can further approximate the helmholtz equation, by basically taking a small angle approximation.
When we do this, we get the paraxial Helmholtz equation, named because it describes light predominantly moving along one axis.
-\frac{\partial^{2} A}{\partial x^{2}}-\frac{\partial^{2} A}{\partial y^{2}}=2 i k \frac{\partial A}{\partial z}
Solutions to this equation show how laser beams change as the propagate through free space.
One of the more popular solutions to this equation is the Gaussian laser beam. What makes this equation nice to use is that it is often easier to solve explicitly than the more complicated, but more fundamental equations seen in coursework.
