So I am taking Calculus 2 at Bowling Green Firelands College right now (Spring 2015) and was looking to take a summer class at Lorain County Community College before I go either Ohio State or U. of Cinci for Electrical Engineering to trim up my curriculum a bit. I am a senior in high school by the way. Originally I was looking to take just Linear Algebra but I noticed that they were offering Diff Eq also at the same school immediately following Linear Algebra.
I kind of figured kill two birds with one stone but when I looked at Diff Eq said that I needed Multivariable Calculus to take it. At other schools I have checked, they only require you to have Calculus 2. Because of that, I do not know whether I will be able to take both or not. I am kind of under the impression that since it is a higher level course and that it is offered over the summer, they are going to throw anyone into it to put it through.
Is there anything from Multivariable Calculus that is used extensively in Diff Eq aside from maybe partial derivatives which are fairly straight forward as is? So should I have any problem taking them concurrently? I know it is going to be a large load.
Why don’t you contact the college to see what their requirements are? They can help you. Also, check with OSU and UC to see if you can take courses over the summer and if they will transfer. Some Us don’t allow it.
Were you awarded any scholarship $$? If you did, make sure you won’t impact your entering freshman status by taking any college level coursework post high school graduation. Some schools will automatically consider you a transfer student if you take classes after high school.
It depends on the DE course. They almost always use functions of >1 variable, which is the point of multivariable calculus. You have to know how to work with them. Most colleges have MV at least as a corequesite to DE because of this. Why can’t you take MV first? You’ll definitely need it anyways.
My school requires multivar as a prerequisite, but I used almost none in my diffE class. If you want to take two math classes together, I suggest taking multivar and diffE concurrently and then LA because I personally think ordinary differential equations was really easy, multivariable calculus a bit harder, and linear algebra the hardest. Linear algebra requires a little more mathematical maturity than the other two, IMHO.
Good luck, fellow math nerd! 
It depends on the level at which the Diff EQ class is being taught. I took an intermediate level differential equations course last semester, after taking calc 2. Now I’m taking calc 3 this semester. Some methods of solving differential equations involve concepts like partial derivatives that don’t get covered until calculus 3. However, the basics of partial derivatives can be picked up very quickly. At this level, the only real prerequisite is calculus 2, because one must be pretty proficient with differentiation and integration.