This question first requires some "sorting".
As long as You want to do that with nitrogen or oxygen, so called "permanent gases"
in this context, the answer is no. Linde machines are plants! The first liquefied air was
made in lab (Krakau) with simpler apparatus, but nevertheless I doubt You could tinker something like that, high pressures and precooling to rather deep temperatures are necessary.
If You content with "not permanent" gases, then just take a flask for refill of lighters
(Butane) and open the nozzle pointing it down into a glass. some liquid and cold and boiling butane will collect.
"dry" ice is simple too, just take a fire extinguisher filled with liquid carbon dioxide
and open the nozzle and blow into a bag made from coarse linen. Dry ice (snow) will collect in the bag. Depending on inner construction of the gas cylinder (raiser or not)
You have to hold it upside down maybe. In general do the way prescribed for the extinguisher in use.
Edit, I forgot Your asking for "equations". Look for "Joule-Thompson-effect" (Wikipedia)
the only thing You will need, is the inversion temperature. Calculations maybe will show You the difficulty to make liquid air.
If You try to tinker in this domain, please be aware that steel (and other metals)
will be very brittle at liquefied air temperatures.(Danger of explosions) Selection of materials is a very important part of cryogenics.