I need to create a search form that will display search results from specific category

You can use WP_query() to narrow down the results of your search:

<?php $args = array(
    's' => $_GET['s'],
    'post_type' => array( 'post', 'page' ),
    'post_status' => 'publish',
    'category_name' => 'music',
    'posts_per_page' => -1
);
$custom_search = new WP_Query( $args );
if ( $custom_search->have_posts() ) {
    while ( $custom_search->have_posts() ) : $custom_search->the_post(); ?>
         <div class="entry-content">
            <h2 class="title"><a href="https://wordpress.stackexchange.com/questions/262512/<?php the_permalink(); ?>"><?php the_title(); ?></a></h2>
            <?php the_excerpt(); ?>
        </div>

    <?php endwhile;

} else { ?>
    <h2>Your search didn't return any results.</h2>
<?php } ?>