Example of a search form, searches with Google in utwente.nl/en/id
(Industrial Design programme website).
How it works:
<form id="search" method="get" action="https://www.google.com/search">
<input type="text" size=20 name="q">
<input type="hidden" name="sitesearch" value="utwente.nl/en/id">
</form>
Alternative: Google Custom Search Engine.
To search inside a page, you can use a jQuery Plugin like Tipue or Searchable.