Cleve Moler has stated that "all numerical methods for stiff odes are implicit." However, I don't know whether this statement is a mathematical fact, or an simply an observation. Moreover, many numerical methods for ODEs assume that the only thing we can do with the right-hand side is point evaluation. If, for example, we can use automatic differentiation on the right-hand side of $x' = f(x,t)$, we are led into Taylor methods for ODEs.
But can Taylor methods be used effectively on stiff equations? If not, are there implicit Taylor methods which patch this problem up?
Note: There is always a problem about the definition of stiff. Will Cleve's definition work in this context?
A problem is stiff if the solution being sought varies slowly, but there are nearby solutions that vary rapidly, so the numerical method must take small steps to obtain satisfactory results.