You can use an if statement in the header
if ( is_home() ) {
<img src="https://wordpress.stackexchange.com/home-image.png"/>
} else {
<img src="/other-image.png"/>
}
Below are a few more way to pick a specific page
// When any single Page is being displayed.
is_page();
// When Page 42 (ID) is being displayed.
is_page( 42 );
// When the Page with a post_title of "Contact" is being displayed.
is_page( 'Contact' );
// When the Page with a post_name (slug) of "about-me" is being displayed.
is_page( 'about-me' );