Does the existence of a college course depend on the department in which it is studied?
The only sure answer to this question is "it depends".  This is a business rule and the organization will set this business rule.

Here is how you can go about thinking about what the business rule might be and how you can determine the best way to model this relationship.

Think first about cardinality.  Is course related to only a single department?  If a course can be multidisciplinary and offered jointly by two departments then you know right away that course is a strong entity.

Second, think about functional dependencies.  What are your candidate keys for course?  Do you have a candidate key that does not make reference to department?  If so, then you have good reason to presume that course is a strong entity.  Do all of the candidate keys of course reference the department that offers it?  That may be a strong indication that course is a weak entity.

Third, think about the future.  Let's assume that you determine from the above that course seems to be a weak entity.  Is it even remotely conceivable that the factors that led you to this conclusion might change?  If so, you might want to consider promoting course to a strong entity and inventing a candidate key that does not rely on department.  This is a matter of opinion and people may have different philosophies about this.  On the one hand, some people feel that making key changes in the database is a big, painful process to be avoided if at all possible.  On the other hand, some people feel strongly that your data model should reflect your current business rules as closely as possible.