How does dy/dt=ry end up with the constant multiple on the right?

I have, from the textbook:

dy/dt = ry

So I solve and get y = e^(rt)+C

However, the book says that y=y_0 * e^(rt), where y_0 is the answer to the initial condition, y(0).

I don't get it. The + C should be what we use to solve for the initial condition; how does y_0 end up in the same term with the exponential?

Thanks