Drupal is really goofy with a lot of the markup it arbitrarily adds.
If you have a page.tpl file in your template's directory, then open it up and add the following code somewhere before where the html tag starts. Make sure it's within php tags :
Then go to the leading body tag (or whatever) and add this to the body classes part:
$isfront = (drupal_is_front_page()) ? " this-is-front" : "";
The drupal_is_front_page() will return true if it's the front page, which will set your $isfront class variable. If not, then your variable will be blank. You're then tacking it on to the rest of the body classes there, and you should be good to go. If it's the front page, then you should have a new class: "this-is-front" to work with.
<body class="<?php print $body_classes.$isfront; ?>">