As per my understanding, you need to get upcoming birthdays in next 5 days. In MySQL
you can do with the query below,
SELECT *, DATE_FORMAT(DateOfBirth,'%m-%d') AS md
FROM Table_A
WHERE DATE_FORMAT(DateOfBirth,'%m-%d') BETWEEN '08-07' AND '08-12'
ORDER BY md
It will return all birthdays between 07 and 12 Aug. Get the current and +5 days at the scripting side.
If you want pure query based solution without any calculation for dates at scripting side, refer following query,
SELECT *, DATE_FORMAT(DateOfBirth,'%m-%d') AS md
FROM Table_A
WHERE DATE_FORMAT(DateOfBirth,'%m-%d') BETWEEN DATE_FORMAT(CURDATE(), '%m-%d')
AND DATE_FORMAT(DATE_ADD(CURDATE(), INTERVAL 5 DAY), '%m-%d')
ORDER BY md