Apollo 13: Problems-

The Apollo 13 malfunction was caused by an explosion and rupture of oxygen tank no. 2 in the service module. The explosion ruptured a line or damaged a valve in the no. 1 oxygen tank, causing it to lose oxygen rapidly. The service module bay no.4 cover was blown off. All oxygen stores were lost within about 3 hours, along with loss of water, electrical power, and use of the propulsion system.
The oxygen tanks were highly insulated spherical tanks which held liquid oxygen with a fill line and heater running down the centre. The no. 2 oxygen tank used in Apollo 13 (North American Rockwell; serial number 10024X-TA0008) had originally been installed in Apollo 10. It was removed from Apollo 10 for modification and during the extraction was dropped 2 inches, slightly jarring an internal fill line. The tank was replaced with another for Apollo 10, and the exterior inspected. The internal fill line was not known to be damaged, and this tank was later installed in Apollo 13.
The oxygen tanks had originally been designed to run off the 28-volt DC power of the command and service modules. However, the tanks were redesigned to also run off the 65-volt DC ground power at Kennedy Space Centre. All components were upgraded to accept 65 volts except the heater thermostatic switches, which were overlooked. These switches were designed to open and turn off the heater when the tank temperature reached 80 degrees F. (Normal temperatures in the tank were -300 to -100 F.)

During pre-flight testing, tank no. 2 showed anomalies and would not empty correctly, possibly due to the damaged fill line. (On the ground, the tanks were emptied by forcing oxygen gas into the tank and forcing the liquid oxygen out, in space there was no need to empty the tanks.) The heaters in the tanks were normally used for very short periods to heat the interior slightly, increasing the pressure to keep the oxygen flowing. It was decided to use the heater to "boil off" the excess oxygen, requiring 8 hours of 65-volt DC power. This probably damaged the thermostatically controlled switches on the heater, designed for only 28 volts. It is believed the switches welded shut, allowing the temperature within the tank to rise locally to over 1000 degrees F. The gauges measuring the temperature inside the tank were designed to measure only to 80 F, so the extreme heating was not noticed. The high temperature emptied the tank, but also resulted in serious damage to the Teflon insulation on the electrical wires to the power fans within the tank.


56 hours into the mission, at about 03:06 UT on 14 April 1970 (10:06 PM, April 13 EST), the power fans were turned on within the tank for the third "cryo-stir" of the mission, a procedure to stir the liquid oxygen inside the tank which would tend to stratify. The exposed fan wires shorted and the teflon insulation caught fire in the pure oxygen environment. This fire rapidly heated and increased the pressure of the oxygen inside the tank, and may have spread along the wires to the electrical conduit in the side of the tank, which weakened and ruptured under the pressure, causing the no. 2 oxygen tank to explode. This damaged the no. 1 tank and parts of the interior of the service module and blew off the bay no. 4 cover.
Solution:
The lunar module had charged batteries and full oxygen tanks for use on the lunar surface, so Kranz directed that the astronauts power up the LM and use it as a "lifeboat"– a scenario anticipated but considered unlikely. Procedures for using the LM in this way had been developed by LM flight controllers after a training simulation for Apollo 10 in which the LM was needed for survival, but could not be powered up in time. Had Apollo 13's accident occurred on the return voyage, with the LM already jettisoned, the astronauts would have died, as they would have following an explosion in lunar orbit, including one while Lovell and Haise walked on the Moon. 
A key decision was the choice of return path. A "direct abort" would use the SM's main engine (the Service Propulsion System or SPS) to return before reaching the Moon. However, the accident could have damaged the SPS, and the fuel cells would have to last at least another hour to meet its power requirements, so Kranz instead decided on a longer route: the spacecraft would swing around the Moon before heading back to Earth. Apollo 13 was on the hybrid trajectory which was to take it to Fra Mauro; it now needed to be brought back to a free return. The LM's Descent Propulsion System (DPS), although not as powerful as the SPS, could do this, but new software for Mission Control's computers needed to be written by technicians as it had never been contemplated that the CSM/LM spacecraft would have to be maneuverer from the LM. As the CM was being shut down, Lovell copied down its guidance system's orientation information and performed hand calculations to transfer it to the LM's guidance system, which had been turned off; at his request Mission Control checked his figures. At 61:29:43.49 the DPS burn of 34.23 seconds took Apollo 13 back to a free return trajectory. 
Challenges during this solution:
1.Life Support Challenges and Carbon Dioxide Scrubber:
Problem: With the oxygen tank explosion, the lunar module was repurposed as a lifeboat, but it had limited resources to support three astronauts for an extended period.
Solution: The team on Earth devised a solution to manage the increasing levels of carbon dioxide in the lunar module. They developed a procedure to construct a makeshift carbon dioxide scrubber using materials available on the spacecraft. The procedure involved fitting square lithium hydroxide canisters, designed for the command module's round connectors, using duct tape and plastic bags. The crew followed these instructions, effectively reducing the levels of carbon dioxide to safe levels and maintaining breathable air.

2.Course Corrections and Safe Re-entry:
Problem: With the loss of propulsion in the service module, Apollo 13 needed a series of precise engine burns to adjust its trajectory for a safe return to Earth.
Solution: Mission control calculated a series of engine burns using the lunar module's descent engine. The crew had to manually input these calculations into the guidance system using paper and pencil. The calculated burns were essential to ensure that the spacecraft would hit the re-entry corridor in Earth's atmosphere, which was a small target from such a distance.

3.Power-up of Command Module and Re-entry:
Problem: The command module had been shut down to conserve power, and it needed to be safely powered up for re-entry.
Solution: Mission control developed a procedure to guide the crew through the process of safely powering up the command module. This included reactivating the command module's systems and ensuring that critical components were operational for re-entry. The crew followed these instructions successfully, allowing the command module to be ready for the re-entry process.