You can use a Child Theme to make modifications to your theme.
Add this to your functions.php file:
function custom_image_sizes() {
add_image_size('home-images', 768, 300, true);
}
add_action( 'after_setup_theme', 'custom_image_sizes' );
You can then make a duplicate copy of the themes’ content.php file and place it in your Child Theme folder, make the following code change to it.
FIND:
<a href="<?php the_permalink(); ?>" title="<?php echo esc_attr( sprintf( __( 'Permalink to %s', 'ryu' ), the_title_attribute( 'echo=0' ) ) ); ?>" rel="<?php the_ID(); ?>" class="ryu-featured-thumbnail">
<?php the_post_thumbnail( 'ryu-featured-thumbnail' ); ?>
</a>
REPLACE IT WITH:
<a href="<?php the_permalink(); ?>" title="<?php echo esc_attr( sprintf( __( 'Permalink to %s', 'ryu' ), the_title_attribute( 'echo=0' ) ) ); ?>" rel="<?php the_ID(); ?>" class="ryu-featured-thumbnail">
<?php if (!is_home()) { ?>
<?php the_post_thumbnail( 'ryu-featured-thumbnail' ); ?>
<?php } else { ?>
<?php the_post_thumbnail( 'home-images' ); ?>
<?php } ?>
</a>
You’ll also want to Regenerate Your Thumbnails to ensure the new image size is being used on the homepage.
Hope this helps.