没有数据库选择错误MYSQL到PDO

Question

function execute($rsql=null) {
global $db_hostname,
$db_username,
$db_password,
$db_name;   
$conn = mysql_connect($db_hostname,$db_username,$db_password) or die ("Error: could not connect to database");
$conn = new PDO('mysql:host=localhost;dbname=$db_hostname', $db_username, $db_password); // I TRIED THIS
mysql_set_charset('utf8');
mysql_select_db($db_name);
if($rsql != null){
  $result = mysql_query($rsql) or die (mysql_error()); 
  return $result;     
  mysql_close($conn);
}else{
  define('conn',$conn);
}

}

大家好，我有这样的错误：当我将$conn = mysql_connect($db_hostname,$db_username,$db_password) or die ("Error: could not connect to database");更改为 PDO 会出现在我的页面中：No database selected.任何人都可以帮助我吗？