The B-side to the great Rock and Roll springtime single from the Beatles in 1969 was an original Lennon/McCartney composition written mainly by John (for Yoko) entitled "Don't Let Me Down". This was one of many love songs written by John with his then-girlfriend Yoko Ono in mind.
The performance of "Don't Let Me Down" that was released commercially on the single was recorded at Apple Studios at 3 Savile Row, London on January 28 1969. There was a false start and then a complete take which was the one chosen for the B side. There were a couple of vocal problems, however, and these were probably fixed sometime in February at Trident but it is not clear when and where. The information concerning this song is still sketchy (as is a lot of the "Get Back/Let It Be" numbers) since the exact takes and rehearsals of each individual song is blurry and not clear in most of the books I've read. There is no doubt, though...both "Get Back" and "Don't Let Me Down" definitely come from the above date.
Another problem is knowing exactly when the mixing was done for the mono version of "Don't Let me Down": The "Get Back/Don't Let Me Down" single was released in stereo in the North American market but was released in mono in the rest of the world (including the UK). The information pertaining to "Get Back" is fairly cohesive although the mono mix for "Don't Let Me Down" can be attributed to April 04 1969 (according to John Winn's book) although there is no location and/or it can be simply a mix-down of the stereo mix conducted at Olympic Studios on April 07 1969. Winn's book also claims that the mono mix runs slightly slower although I haven't compared them.
"Don't Let Me Down" can be found on the B-side of the Apple 45 and CD single R5777 (in the UK) and 2490 ( in North America) as well as being on the Apple LP compilation "Hey Jude/The Beatles Again" . It is also on the Apple 2-LP/2-CD "The Beatles 1967-1970". It can be found on the EMI/Capitol 2-LP/CD "Past Master Volume 2" as well as on the Apple/EMI 2-CD "Mono Masters" from the mono box set and the Apple/EMI 2-CD "Past Masters". There is a different mix of both sides of the single available on the Apple/EMI 2-CD/ LP/45 "Let It Be...Naked".