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The OED says it's "after German schwanen(ge)sang, schwanenlied".

Being the OED, they're probably right. They give the meaning as:

a song like that fabled to be sung by a dying swan; the last work of a poet or musician, composed shortly before his death; hence, any final performance, action, or effort.

"swan, n." OED Online. Oxford University Press, September 2014. Web. 24 November 2014.

Michael Quinion goes into more detail on World Wide Words:

An ancient legend — it goes back to classical Greece — holds that swans are silent throughout their lives but sing once, beautifully, just before they die. ...

The legend is all nonsense, of course. The ancient Greeks and Romans believed it, however, and it’s mentioned in the works of Euripides, Aristotle, Plato, Seneca, and Cicero. Plato said that Socrates had explained it as a song of gladness because the swan, sacred to the god Apollo, was shortly to join the god it served.

In AD77 Pliny said it was untrue (“olorum morte narratur flebilis cantus, falso, ut arbitror, aliquot experimentis”, “observation shows that the story that the dying swan sings is false”). He had no effect on the popularity of the fable; the idea was eventually taken into English in the medieval period, being alluded to by Chaucer and Shakespeare. It’s in the latter’s The Merchant of Venice: “Let music sound while he doth make his choice; / Then, if he lose, he makes a swan-like end, / Fading in music.”

But the term itself was created as recently as 1831, in a book by Thomas Carlyle: “The Phoenix soars aloft ... or, as now, she sinks, and with spheral [sphere-like] swan-song immolates herself in flame.” He took it from the German Schwanenlied or Schwanengesang with the same sense, which derives of course from the same legend. As the final collection of songs by Franz Schubert was published in the year he died (1828), it is known as his Schwanengesang. That was probably what put the idea for the English word in Carlyle’s mind.

World Wide Words: Swan Song

Michael has written a number of books on the English language for Oxford University Press and Penguin, and is an adviser for the OED. I'd put quite a lot of faith in his website.

Although MQ is likely to be right about the etymology, he may be wrong when he says that the legend itself is 'all nonsense'. Although the mute swan (which is the kind of swan that is most common in Europe) definitely does not make any special vocalisations as it dies, the noisy Whooper Swan (C. cygnus cygnus or cygnus musicus or Olor cygnus) aka 'whistling swan' may do so; see Bryan P. Martin, Wildfowl of the British Isles and North-West Europe, for further info on the swan itself.

Here is what purports to be an eye-witness account of a 'stricken' swan's song:

Wings fixed, he commenced at once his song which was continued until the water was reached nearly a mile away. Never before or since have I heard anything like the song of this stricken swan. It sounded at times like the running notes of an octave, most plaintive in character and musical in tone and, as the sound was borne to us, mellowed by the distance, we stood astounded and could only exclaim, "We have heard the song of a dying swan".

H. W. Robinson, quoted in:
D'Arcy Thompson, A Glossary of Greek Birds (London and Oxford 1936), p.183, cited in:
Amber, Avallon, and Apollo's Singing Swan
Frederick M. Ahl
The American Journal of Philology, Vol. 103, No. 4 (Winter, 1982), pp. 373-411
Published by: The Johns Hopkins University Press
Article DOI: 10.2307/294518
Article Stable URL: http://www.jstor.org/stable/294518

The classicist W. Geoffrey Arnott goes into more detail:

the whooper, unlike the mute, has a remarkably shaped trachea, convoluted inside its breastbone;" and when it dies, the final expiration of air from its collapsing lungs produces a 'wailing, flute-like sound given out quite slowly'. In modern times this dirge of the dying whooper was first attested by the great ornithologist Peter Pallas in Russia at the beginning of the nineteenth century. It has been more recently observed also in other species of wild swan with similarly convoluted tracheae. The American ornithologist Dr. Daniel Elliott once shot a whistling swan (Cygnus columbianus, the American subspecies of the Eurasian Bewick's swan) for the American Museum of Natural History, 'and as the bird came sailing down he was amazed to hear a plaintive and musical song, so unlike the call in life, which lasted until the bird reached the water.'

Swan Songs
W. Geoffrey Arnott
Greece & Rome, Second Series, Vol. 24, No. 2 (Oct., 1977), pp. 149-153
Published by: Cambridge University Press on behalf of The Classical Association
Article Stable URL: http://www.jstor.org/stable/642700
(emphasis mine)

Arnott adds:

one pleasant irony remains. Plato's account of the legend at Phaedo 84 e-85 a, which has been pilloried as a fanciful fiction by most scholars from Pliny and Alexander of Myndos through the ages right up to the present day (Scaliger, for example, dismissed it as an audacious invention of 'Greece, parent of falsehood'), turns out to be precisely accurate in all its details.

(One should perhaps note that as a mid-20th-century British classicist, Arnott may have something of a vested interest in supporting Plato over the "Greece, parent of all falsehood" view).

For more info see:

• Daniel Elliott, "plaintive and musical song", is quoted in Swans of the World, Sylvia Bruce Wilmore, Taplinger Pub. Co., 1979, p.129. ISBN 9780800875237.

• "wailing, flutelike sound": Practical Handbook of British Birds, Witherby, 1919. This quote from 1940 edition, iii.169.

• Zoographia rosso-asiatica, Pallas, 1831 (on Google Books but written in Latin).

I will make up some sentences with context so you can get a feel for these.

I worked on this assignment for three hours and then the professor sent an email saying there was a serious error in the assignment and it was cancelled. Jeez!

"Work at" something implies persistence, chipping away at something.

I'll tell you how I managed to submit this assignment on time. I worked at it for a couple hours every day and then did an all-nighter on the last day. Whew!