Short for no patient:
RUN SQL query against GDB_ITEMS table, find the row has that table name, feature dataset is in the path field
Long story:
I know the table name in SQL server. But I can't find it in ArcSDE
Why? because in ArcSDE all you see is the feature dataset table, not the table name. Table name is same as feature layer name.
You have to open each feature dataset, it includes multiple feature layer. Feature layer name is SQL table name, it just hide inside feature dataset. I have hundred of feature dataset, I don't know this SQL table( feature layer) is under which feature dataset. There is no search function let you quick find out.
GDB_ITMES table has relationship between feature dataset and each feature layers(same as SQL table name)
RUN SQL query, find the row has that table name, feature dataset is in the path field