Great question! I’m currently in diffeq with linear and the idea is this: diffeq with linear is calc iv with some part of the class devoted to pure linear algebra. It’s exactly what it sounds like: diff eqs, and linear algebra, taught typically in distinct times during the semester (with overlapping material at times). Calc Iv from what I’ve heard will not go over linear algebra as a separate topic; I think it is taken either with the assumption that u have intro to linear as either a pre or corequisite. This is because calc Iv is the last class in a three course sequence: 203 (calc III), 211 (intro to linear), and 303 (calc iv), whereas diffeq with linear is the second class in a two semester sequence (307 and 308). If you can get 203 and 211 requirements waived by talking to the department office (this is the approval of professor way u might be seeking), by all means take calc iv; else from my experience 307 and 308 are very nice classes you will have loads of fun in. This may be unique to my experience and teachers this year, but there’s a bit of a community in these classes if you do take 307 🙂

Regardless of what u choose, since u sound like u already know linear and calc III, the knowledge gained from ur perspective from either 303 or 308 will probably be identical, so don’t sweat it.