Display just child pages of a certain page in search results

Try the following approach.

Get the page using get_page_by_path()

http://codex.wordpress.org/Function_Reference/get_page_by_path

$page = get_page_by_path( 'university-guide' );

$search_query['post_type'] = 'page';
$search_query['post_parent'] = $page->ID;

The following snippet was taken from Codex page: Creating a search form
http://codex.wordpress.org/Creating_a_Search_Page

global $query_string;

$query_args = explode("&", $query_string);
$search_query = array();

foreach($query_args as $key => $string) {
    $query_split = explode("=", $string);
    $search_query[$query_split[0]] = urldecode($query_split[1]);
} // foreach

$search = new WP_Query($search_query);

Search results are obtained using WP_Query instance.

You can use WP_Query codex page to learn more about available options.
https://codex.wordpress.org/Class_Reference/WP_Query