It’s impossible to say for sure without knowing the value of $start_date and $end_date. It may be they’re the wrong types; PHP would expect a string here. You can find out what type you’re dealing with by using var_dump($start_date, $end_date); to print the values out.
It’s also worth noting that sticking values into an SQL statement like this will make your code insecure; look into using prepared statements to prevent this. Here and here are good resources for learning about prepared statements, using the preferred PDO system. The rest of these sites are useful for learning the rest of PHP as well.
When you submit a form to PHP, the user’s input is available in the $_POST global array. The array keys will correspond to the name attribute of the input tags in your form, rather than the id attribute you seem to be expecting.
If you var_dump($_POST) in your submit.php code, you should see something like the following:
You can access these values using $_POST['start'] and $_POST['end']. Your user isn’t guaranteed to fill them out correctly, so you should check that the values are present and look like you expect.