When page_load fire on page after any event then dropdownlist2 is not visible. so check through condition under :
protected void Page_Load(object sender, EventArgs e)
{
if (dropdownlist2.SelectedIndex!=0)
dropdownlist2.visible=true;
else
dropdownlist2.visible=false;
}
go through this. Hope this will help you.
Don't forget to mark as answer if it helps. :)