I’ve come across this problem in my profession and my solution is much simpler and easier to understand than the one proposed in this verbose write-up.

#!/bin/bash
if [ `date --date="+1 day" +%d` = 01 ]; then
execute desired task
fi

Have cron execute this bash script at the desired run-time each day to achieve the desired results.

Plug. Chug. Done.

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