MIT, for example, doesn't require their CS students to take a differential equations class. So you're wrong in that respect. The rest is certainly doable though.
There's nothing in this exam that requires specific diff eq knowledge.
Doesn't question 2, problem 2?
No? That's calc 2 level problem, not a diff eq one. Partial derivatives are hardly diff eq territory.
Sorry, I'm not american. What does calc 2 normally refer to in the US?
Though if you've taken a course on variational calculus then question 2 is completely routine and you're at a great advantage relative to someone who hasn't taken such a course.