Use DateTime::createFromFormat like this :
$date = DateTime::createFromFormat('m/d/Y H:i:s', $input_string.' 00:00:00');
$mysql_date_string = $date->format('Y-m-d H:i:s');
You can adapt this to any input format, whereas strtotime() will assume you're using the US date format if you use /, even if you're not.
The added 00:00:00 is because createFromFormat will use the current date to fill missing data, ie : it will take the current hour:min:sec and not 00:00:00 if you don't precise it.