They had a custom textbook created for their 2-course ODE sequence that several of the faculty collaborated on. Though it did contain content on uniqueness theorems and some proofs, far and away the biggest two items hammered in were (a) linear equations with constant coefficients, and (b) Laplace transform methods.
They also offered (at the time) a 3rd, optional course called Boundary Value Problems that was focused on several physics-motivated BVPs like with Laplace's equation, heat equation, wave equation, Young's modulus, and others, and that course heavily used Fourier and Laplace methods.
We did have word problems, but they were almost exclusively "salt tank" problems. Literally, every word problem described a tank of water or pre-mixed brine solution, with some description of either more salt or more water being added or removed, either gradually or in discrete injections.
The fact that every problem was an infamous "salt-tank problem" essentially made its status as a word problem irrelevant. This seems like it wouldn't be that helpful but actually it was really nice. You got so used to the different pieces that comprised the modeling problem that when you went off and did something in other courses, like circuit systems or conservation systems in mechanical engineering, you knew how to translate the problem to 'salt tank' form, which really covered a huge range of practical problems.
As a math major, one fault I noticed of this method was that it did not make the connections to linear algebra very clear. It took me another few semesters afterward to catch up on that part, but I can understand how engineering majors cared less about that.
I don't know what Rose-Hulman does for this curriculum now, but it would be cool to somehow take a "snapshot" of their methods for it and compare it with other experiences like this OP.