To expand on the answer a bit. So for 2-dimensional first order ODEs there is the Poincaré–Bendixson theorem. This tells you all the possible types of solutions that can occur. Using this and studying the vector field it is easy to prove which solutions occur.
In the case of equilibria (for n-dimensional ODEs) you have the Stable Unstable and Center Manifold Theorem. These theorems tell you what happens close to an equilibrium.