No, typically things end at single variable calculus, although it depends on the school. I can't imagine that there is much value to having high school students doing diff eq.
At Caltech specifically FWIW, the biology major requirements include intro to diff eq (Ma 2),[0] which is usually taken in the first quarter of sophomore year.
yes but caltech is atypical (at many schools, biologists would fail diff eq just like many biologists fail organic chemistry). This course is definitely designed for educated, motivated students.