What are the elements of literary analysis present in "Dog's Death" by John Updike?

I will discuss each of the elements of literary analysis and how they are applied or are relevant in the poem "Dog's Death."

The first element is setting, which relates to the time, place, circumstances, conditions, and atmosphere in which a literary piece is set. In the poem, the physical setting shifts from the speaker's home to his car and then to the veterinarian. In each of the settings referred to in this regard, the atmosphere and tone are specifically mentioned. It is clear that the dog had been a much-loved companion "surrounded by love" at home, and she is described in warm tones, making the initial mood and tone bright and optimistic. The dog does well in following instructions and is complimented: "Good dog! Good dog!" Her death is poignantly depicted and the mood is sad and quite depressing in the journey to the veterinarian when she dies. Even the family's love could not save her:

Though surrounded by love that would have upheld her,
Nevertheless she sank and, stiffening, disappeared.

The second element is character: The dog is obviously the main character since she is the subject of the poem. She was a young pup in training, much loved by the family and also intelligent and obedient, for she quickly learned how to use the newspaper to relieve herself. It is quite poignant that even when the dog was bleeding that she tried to do so on the paper, as she had learned. The pup is the focus in the poem and the family is of secondary importance, although their sentiments are an important element of the drama which unfolds in the poem.

Important with regard to character is also point of view, since the speaker in the poem is also involved in the events depicted. In this instance, the speaker is the father of the family who tragically lost their dog. He shares his sentiments, and it is clear that he was quite distraught. Also evident is the fact that he was the one who tended to the puppy in her time of desperate need and, with his wife, took the dog to the vet. Her injury, "a rupture in her liver," was much too severe to do anything about. 

Another element is the use of imagery. The poem does not have much imagery since the narrative style is quite straightforward. The poet has, however, employed an effective metaphor in line 8:

And her heart was learning to lie down forever.

On a deeper level, one could also argue that the poem is a symbol for man's contradictory relationships with the animals with whom we share the planet. On the one hand, we have those who care and on the other, those who are abusive and uncaring. The first line states: 

She must have been kicked unseen or brushed by a car.

The puppy's death is the result of man's intervention, for it could only have been a human who could have kicked her or driven the car which caused her injury.

Some alliteration has also been used for emphasis, such as in line nine, which begins with, "Monday morning," which emphasizes the day on which this tragic event occurred.

The poem has a fairly simple structure and no definitive rhyme scheme. The rhyme is quite disjointed, with some lines rhyming and others not. This is probably done to indicate the unusual nature of the event being described. The young pup's death is unexpected and unwarranted and not even the family's best efforts can save her. The simple structure of the poem furthermore indicates that although the dog's death is unusual and surprising, everything else is not. The poem is about an ordinary family owning an ordinary pet.

The theme is loss. The poem conveys the speaker and his family's grief at losing a companion, something none of them foresee happening so soon. The family had grown attached to this obedient friend. The speaker emphasizes the dog's tractable and pleasing nature by repeating the phrase, "Good dog." It is tragically poignant that she remained good to the end. The poet effectively conveys this by ending the poem with the same phrase.