TRY this :
$db_user = "username";
$db_pass = "password";
exec("mysqldump --opt -u$db_user -p$db_pass --no-create-info --where date >= '2013-08-01' my_database my_table > backup.sql");
TRY this :
$db_user = "username";
$db_pass = "password";
exec("mysqldump --opt -u$db_user -p$db_pass --no-create-info --where date >= '2013-08-01' my_database my_table > backup.sql");