Typically, dual enrollment (DE) programs are set up by the school board/school district where your high school is located and there are written agreement between the colleges and the school board/district. In most cases (not all!), the students does not incur any cost (tuition/fees/books) and there are requirements that are required to be met in order to qualify as a dual enrollment student (minimun GPA, SAT/ACT scores or results on entrance exams, etc.).
You should speak with your guidance counselor to see if your high school have a DE program in place and what are their requirements. Some school districts partner with local community colleges and/or 4-year institutions (written agreement between the local colleges and the school district). Your DE courses would technically become part of your high school record/transcript and affects your GPA and also, these courses can be used to satisfy your HS graduation requirements.
Have you taken AP Calculus BC? Calculus BC would be a prerequisite for Calculus III (multivariable calculus) and differential equation prerequisite is Calculus III, while differential equation is a prerequisite for linear algebra/vector analysis.