how to backup with this connection string

If you can connect to the database using this string in order to view records, then you should also be able to backup the database: Since you don't specify how you are doing that we can't debug that part for you, but there is an example here: http://www.geekpedia.com/tutorial180_Backup-and-restore-SQL-databases.html[^]

i am sure that this below link will help you a lot.
LINK
just read and follow instruction.

here is code spinet.

To BACKUP and RESTORE DataBase through SQLCMD.EXE :
PATH OF EXE:- C:\Program Files\Microsoft SQL Server\100\Tools\binn\SQLCMD.EXE

sqlcmd -S.\SQLExpress
1> BACKUP DATABASE dbName TO DISK = 'path'
2> GO
An alternative, single-line backup command:

sqlcmd -S.\SQLExpress -Q"BACKUP DATABASE dbName  TO DISK = 'path'"
(Note that you must enclose the path within quotes)

To Restore:

sqlcmd -S.\SQLExpress
1> RESTORE DATABASE dbName FROM DISK = 'path'
2> GO

For Restore with REPLACING EXISTING DATABASE
1> RESTORE DATABASE dbName FROM DISK = 'path'
2> WITH REPLACE
3> GO

I wonder if you really mean backup using C# with the question, but if you do, then you:
- open the connection with working connection string
- create a SqlCommand object
- define a backup command in T-SQL
- execute the command using ExecuteNonQuery
- handle possible errors
- close the connection
- dispose the connection

And the backup command itself: BACKUP (Transact-SQL)[^]