general solution for differential equations

Hi I wasn't sure whether to post this in the linear algebra forum or the calculus forum, but it deals with differential equations. I'm having a hard time understanding this one question and how it differs from the normal case. The question is:

Find the general real-valued solution for the system of equations:

Now, I know how to solve equations of the form

where f is just some function or constant. The confusing part for this problem is that & are in both equations, so you can't just solve each equation outright like you typically would when there is just y and not the y1 and y2.

Once the general equation is known, I know its pretty easy to solve the initial value problem I need to solve after, but I just need help on how to approach the problem.

Am I supposed to somehow substitute to get one equation with all y1 and the other with all y2 or how would I go about doing it? It just seems like a weird problem to me that I've never encountered before.