You say that you are not sure what "analytical and numerical methods" means, so let me try to fill you in on that part.
Analytical methods refers to the methods you probably learned when you too DE in college, such as separation of variables which you mention, the use of an integrating factor which you also mention, probably Laplace transforms, maybe Fourier transforms, perhaps some stuff on Bessel functions, Legendre functions, Hankle function, and elliptic functions and the ODEs associated with each type of function. There is plenty of material of this sort to go for many semesters, so it is hard to say exactly what you will get, but it will be some of this.
Numerical methods means ways of doing arithmetic to obtain answers to various sorts of problems. For example, the numerical solution of linear algebraic equations is a large area of study involving ways to solve systems of algebraic equations by means of the computer, where the computer does nothing more than a lot of arithmetic, but it does it very fast and with a lot more accuracy than most people. There are also numerical approximations to things like integration, and perhaps most iteresting for your course, the numerical solution of differential equations.
The numerical solution of ODEs has been almost entirely given to IVP problems, with only minimum attention to two point boundary value problems. There is also the very large class of PDEs to be solved, and there are many very effective PDE solutions that are emerging using finite elements and boundary elements. You should be hoping to get at least some of these last items. This is where the new, exciting stuf is today.
Using finite elements and boundary elements it is now possible to solve stress, heat transfer, and electromagnetic field problems in 3D in real geometries. This means solving the actual PDE equations with the proper boundary conditions in these complicated domains. This is really pretty gee whiz stuff.