I have read that ODEs are used in game development, particularly in physics engines and programming

I've used Newton's 2nd and 3rd Laws of Motion a lot, as well as derived equations - i.e., F = ma. I also use torque in several apps - i.e., torque = dL/dt (L = angular momentum). Both used primarily in simulation of motion for objects - reaction to collisions, calculating position/velocity/rotation, etc.

e.g.,

F = m * a

xt = x0 + v * t

I use partial diff. eq. in shaders to find normals - i.e., float3 normal = normalize(cross(ddx(input.worldPos), ddy(input.worldPos))); Very useful HLSL intrinsic functions.

have they studied them ... ?

Oh, yeah. Successful completion of several courses in both [EDIT: oridinary??] ordinary and partial diff-e-q were required for my college degree.

Are ordinary and partial diff-e-q useful? Yes.

EDIT: Occasionally it's convenient to remember, in addition to other similar trig relationships, that cosine(angle) = d(sine) / d(angle) when rate-of-change (slope) is needed.