Problem?
The problem with Apollo 13 was that one of the air tanks exploded, while Apollo 13 was 200,000 miles from earth The air tank's explosion crippled Apollo 13 by stopping its ability to generate electricity, give oxygen, and even make water.
This is a Lithium Hydroxide canister witch purifies the air by absorbing Carbon Dioxide.
Fixing the problem
After the explosion the crew was forced to move to a small detachable part of the ship called the "Lunar Module lifeboat" or "LM lifeboat". The idea was to get the crew in the life boat and use the moon's gravity to slingshot back to earth. NASA fixed the oxygen/carbon dioxide problem by telling the crew to attach Apollo 13's carbon dioxide scrubber to the lifeboat, using a system of tape, plastic bags and cardboard.
The crew fixing one of the LiOH canisters.
getting the crew home
Even then trip back home was dangerous and hard on the crew, dehydration and lack of food plagued the crew causing the crew to loose a total of 31.5 pounds all together. The crew even had to correct their course twice despite the almost none existent energy supply, but the moment of truth was upon reentry to Earth's atmosphere where communication with Apollo 13 went dead silent for a while. After about 4 and 30 seconds of communications black out Apollo 13 reentered Earth's atmosphere and landed in the Pacific ocean on April 17, 1970.