Which course is harder for students? Complex variables designed for engineers/ scientists
odes+linear algebra course for Engineers/Scientists?
Totally guessing but I think the specific ODE course might be a little harder since it is already a big topic without matrices and then you add in LA (so a littler tougher than just a classical ODE course).
That said, I would do the ODE course first. It is much more fundamental than contour integration is to engineering. Almost everything is a diffyQ in engineering. Use of complex variables is actually rather rare in normal engineering homework.
(By use complex variables, I mean full on complex analysis. Of course you use imaginary numbers and phase angles in EE circuit analysis. But that is more like basic algebra 2 type stuff.)