As mentioned in the last song article of Fare Thee Well (Dink’s Song), the song which concludes the Inside Llewyn Davis soundtrack and movie is Bob Dylan’s Farewell. The version below which appears in the movie was released from the Witmark Demos Bootleg Series Vol 9. So I have presented the Witmark Demo from the authorised Dylan channel and the scene from the movie in which a young Dylan appears.
“Farewell“, also known as “Fare Thee Well” was considered for his third album The Times They Are a-Changin’, but Dylan only attempted a few takes during the album’s first studio session. Their are other versions which exist online believed to have taken place in Greenwich Village venues like the basement of either Gerde’s Folk City or the Gaslight Cafe. His friend Happy Traum backed him on vocals and banjo. This bootleg is known as the banjo tape.
According to wikipedia: Dylan based the song on the traditional British folk ballad “Leaving of Liverpool”. He first played it for friends in Greenwich Village after returning from a two-week trip to London in early January 1963. In “Leaving of Liverpool”, the ballad’s first verse and chorus tell the tale of someone sailing from Liverpool to California, bound to miss the loved one left behind.
Farewell to you, my own true love;
I am going far away.
I am bound for Californ-i-a,
And I know that I’ll return someday.So fare thee well, my own true love,
And when I return, united we will be.
It’s not the leavin’ of Liverpool that grieves me,
But, my darling, when I think of thee
“The more I live, the more I learn. The more I learn, the more I realize, the less I know.”- Michel Legrand
Observation Blogger 
We used to sing the original folk ballad at school in the ’50s and also at party “sing-a-longs”! – It was called “Fare thee well”!
I imagine, you guys sang the original English ballad “Leaving of Liverpool”. Nice Bruce you have such a resonating connection with it.
Great movie, great song!