I am a bit confused on the new bootstrap version since they changed dropdown menus to divs:
Neither of the top solutions worked for me.
This works perfectly, keeps submenus open while browsing, add uses the native Bootstrap javascript.
// Mouse over
$('body').on('mouseover', '.dropdown', function(e) {
$(this).children('.dropdown-toggle').dropdown('show');
});
// Mouse leave
$('body').on('mouseleave', '.dropdown', function(e) {
$(this).children('.dropdown-toggle').dropdown('hide');
});