<?PHP
//.. your codes like connection & ...
// my_table is your table name.
$query = mysql_query("SELECT * FROM my_table");
$stored = array();
$i = 0;
while($row = mysql_fetch_assoc($query))
{
    $stored[++$i] = date('Y-m-d H:i:s', $row["start_date"]);
}
mysql_query("alter table my_table modify start_date DATETIME;");
$query = mysql_query("SELECT * FROM my_table");
$i = 0;
while($row = mysql_fetch_assoc($query))
{
    // Id is your primary key.
    mysql_query("UPDATE my_table SET start_date='" . $stored[++$i] . "' WHERE id='" . $row['id'] . "'")
}