Slightly different to all the other but I think this is the most simple:
$(document).ready(function(){
var $this, i, filter,
$input = $('#my_other_id'),
$options = $('#my_id').find('option');
$input.keyup(function(){
filter = $(this).val();
i = 1;
$options.each(function(){
$this = $(this);
$this.removeAttr('selected');
if ($this.text().indexOf(filter) != -1) {
$this.show();
if(i == 1){
$this.attr('selected', 'selected');
}
i++;
} else {
$this.hide();
}
});
});
});