Discourse analysis is defined as (a) concerned with language use beyond the boundaries of a sentence/utterance, (b) concerned with the interrelationships between language and society and (c) as concerned with the interactive or dialogic properties of everyday communication. This is a quotation from M. Stubbs' textbook (Stubbs 1983:1).
Conversation analysis (CA) offers an approach to discourse that has been extensively articulated by sociologists, beginning with Harold Garfinkel who developed the approach known as ethnomethodology , and then applied specifically to conversation. CA differs from other branches of sociology because rather than analyzing social order, it seeks to discover the methods by which members of a society produce a sense of social order. CA is like interactional sociolinguistics in its concern with the problem of social order, and how language both creates and is created by social context. CA provides its own assumptions, its own methodology, including its own terminology, and its own way of theorizing.
CA is a structural methodology for the analysis of talk.
In the first conversation (C1), Mrs Smith is being open and frank with the doctor by explaining the condition of her son. She said that initially she thought that her son was pretending to be sick so that he does not go to school. But when she saw him vomiting and having diarrhea, she realized that he is really sick. Then she took the opportunity to explain her condition as well while she was there. It seems that she did not make a prior appointment for herself.
I think in some cases some doctors may refuse to examine her because they may be pressed for time and they would ask her to make another appointment, this is part of the Australian culture, i.e. the necessity of making appointments to meet other professionals.