There are lots of other methods for solving ODE - as many as you can dream of. Adams-Bashford and Adams-moulton are among the most common These so-called multi-step methods use the current information as well as earlier values of the solution to generate new values - in effect, a higher-order extrapolation for the ODE. Adams-Bashford are explicit, Adams-Moulton implicit.
I personally am partial to Backward Difference Formulae methods, because of their good stability properties. The first-order version of a BDF is backward Euler. A second-order may be written
HOMEWORK: Prove this is a second-order accurate method.
One will note obviously that all these methods generalize directly to systems of ODEs.