Using "did" as an auxiliary like this is (in my experience) very rare. It cannot simply be used in the same places as "didn't", because the "did" there is really just a helper for the "not".
As others have pointed out, the main use of "did" is to contradict a previous statement. This is usually a direct use of "didn't" + verb, or a similar construction like "wouldn't" + verb.
Even then, often the verb is omitted, as in:
I didn't expect to enjoy it, but I did.
The same would be true if two people were talking about their trip:
Alice: I didn't experience any earthquakes during my trip.
Bob: Oh, I did.
Alice: I didn't enjoy the earthquake.
Bob: I did.
The verb would only be included if there was more than one verb that "did" might otherwise apply to:
Alice: I experienced two earthquakes during my trip, and didn't enjoy them.
Bob: Oh, I only experienced one, and I did enjoy it.
A single sentence example, still employing a contradiction and two different verbs:
I didn't enjoy it, but I did find it fascinating.