Compare between the Lilliputians and Brobdingnagians.

The Lilliputians are relatively warlike. They are quite violent. Consider the way they treat Gulliver when they first find him: they automatically assume that he is a threat, and they tie him down, later shooting him with arrows. Not long after, they tie him up and confine him. Further, there have been a high number of civil wars over completely ridiculous things—like which end of the egg is the correct end to crack—that have resulted in the deaths of thousands of Lilliputians. The emperor insists that Gulliver assist Lilliput in its war with Blefuscu, as the emperor wishes to enslave the Blefuscudians.
The Brobdingnagians are relatively peace-loving. When the King of Brobdingnag hears about the invention of gunpowder, he is absolutely shocked by how barbaric and savage Gulliver's countrymen are. He calls Gulliver just about every terrible name in the book, and he refers to Gulliver's race of men "odious little vermin." The Brobdingnagians have few laws, and those laws are relatively simple and straightforward.

When is sarcasm used in Chapter I of The Scarlet Pimpernel?

Sarcasm is employed in the narration of Chapter I with the discussion of the attempted escape from France by the aristocrats. [Examples are italicized.]
The author, Baronness Emmuska Orczy, an aristocrat herself, describes with sarcasm the macabre enjoyment of the people who watch as the aristocrats, who  have been labeled as "traitors to the people" after the Revolution, attempt to flee the country and escape the guillotine. This treatment of the sadistic enjoyment of the peasants is certainly ironic (sarcastic) and intended by Orczy to mock the formerly oppressed peasants, who now themselves become the oppressors.
The "fun" begins when Sergeant Bibot, who has a "wonderful nose for scenting an aristo in the most perfect disguise," does not immediately arrest the aristocrats, but instead toys with them as a cat often plays with a mouse before killing and eating it. This "keen sense of humor" amuses the crowd who stand near the barricades and watch as Bibot often lets his "prey" past the gates in order to make them think they have made their escape. Soon, however, the aristocrat is marched back. To the people this is "extremely funny" because, often as not, the prisoner is a woman, "some proud marchioness, who looked terribly comical" as she has been stripped of her disguise and realizes that she will receive but a summary trial and after this, she will feel "the fond embrace of Madame la Guillotine."
Every day Bibot has the distinct "satisfaction" of apprehending royalists and returning them to the Committee of Public Safety that is presided over by "that good patriot" Citoyen [Citizen] Foucquier-Tinville. But, there is one man who somehow slips past the watchful eyes of the guards. He is an Englishman, the "accursed Scarlet Pimpernel" who cleverly escapes by disguising himself in the most unsuspected personas. Boasting that he would never be so foolish as to let this Englishman go past him, Bibot tells the crowd of one daring escape which cost a guard the punishment of the guillotine. It seems there was a daring escape in which the Scarlet Pimpernel was dressed as a guard and the aristocrats as soldiers while a harmless cart passed out of the city. Hearing this, the crowd becomes silent.

The story savored of the supernatural, and though the Republic had abolished God, it had not quite succeeded in killing the fear of the supernatural in the hearts of the people.

But, further sarcasm comes at the expense of Bibot, who has previously ridiculed the guard deceived by the Scarlet Pimpernel. For, it seems that Bibot has let pass a cart driven by "an old hag" who has told him that her son has had the plague so that he would not wish to be near her or the cart. Ironically, this cart has been allowed by Bibot to pass through with aristocrats hidden in it. And, the "old hag" has been none other than the "accursed Englishman" himself.

What are the shared strategies for W. E. B. Du Bois and Booker T. Washington?

In his chapter on Booker T. Washington in The Souls of Black Folk, Du Bois offers some measured praise of Washington, pointing to places where they agree. He notes that Washington has, as times, sent letters to southern legislatures protesting injustice against black people and that Washington has openly opposed lynching. Du Bois also agrees with him that blacks must "strive mightily" to improve their own lives. He concurs with Washington's emphasis on thrift and industrial education for black people and admires him for stepping up to the plate and taking leadership in the black culture by trying to find a way to make lives economically better for black people. Du Bois also took on a black leadership position and, like Washington, publicly protested lynchings and other forms of racism.
It is important to note, however, that despite a few similarities, the two men differed profoundly in outlook and strategy. Du Bois was sharply critical of Washington. As he outlines in The Souls of Black Folks, Du Bois rejects Washington's strategy of accepting black inferiority and second-class citizenship in return for modest economic gains. Du Bois believes that this strategy is not only ineffective but corrodes and destroys the black soul. As he writes of Washington's acceptance of subservience to whites:

We have no right to sit silently by while the inevitable seeds are sown for a harvest of disaster to our children, black and white.

Booker T. Washington and W.E.B. Du Bois had some shared strategies, even though they later disagreed on the direction of the struggle for equality for African Americans. Both men worked to develop legal strategies to fight the injustices that African Americans faced. Both men also worked together to arrange conferences, both professional and academic. They also wrote essays in the same publications. They also organized the National Negro Business League. They strongly supported the development of African American businesses.
Their goals were to help African Americans deal with the injustices they faced while improving their lives. For a time, both men blamed African Americans for their condition. They believed that self-help and moral improvement were very important qualities for African Americans to pursue. They both believed African Americans should work for economic advancement. They also believed in racial solidarity and working together on economic issues. They were willing to accept voting restrictions based on educational and property qualifications, but not on racial qualifications. They gave economic advancement a higher priority over suffrage rights for African Americans.
While they later differed sharply on how to obtain equality, there was a time when both men shared similar strategies to help African Americans achieve equality and improve their living situations.
http://teachersinstitute.yale.edu/curriculum/units/1978/2/78.02.02.x.html

http://www.alabamahumanities.org/programs/road/presentations/booker-bois/

What are two examples of personification in "Rules of the Game"?

Personification is the use of figurative language in which something nonhuman is described using human traits or characteristics. This is similar to anthropomorphism. The main difference is that, with anthropomorphism, the nonhuman entity (animal, object, idea, etc.) actually takes on the human attribute. (A good example of this is in Animal Farm, in which the animals actually speak.)
The wind is the key symbolic element being personified in this story. Note that at the beginning, we get the first lesson from Waverly's mother:

Strongest wind cannot be seen.

Waverly will learn to be silent and strong like this wind.
As mentioned in a previous answer, the wind whispers secrets only Waverly can hear. In this, her first tournament, she wins her first match. After saying Check, "the wind roared with laughter." The wind is personified as laughing with Waverly's victory. This is the same wind whispering chess secrets and strategies to her. The wind is on her side.
At the end of the story, Waverly goes to her room and imagines a chessboard. Her mother sits across from her and utters the same phrase:

Strongest wind cannot be seen.

Waverly personifies her white chess pieces:

My white pieces screamed as they scurried and fell off the board one by one.

As her mother's chess pieces continue to advance, Waverly feels like she's losing. Then she notes feeling lighter and is pushed up by the wind away from everything below. She uses the wind to escape, a metaphorical tactic she might use in chess or with her mother in the future. The story ends with Waverly pondering her next move, and this undoubtedly refers to her mental chess game with her mother.

Personification is a literary device in which an inanimate object, animal, or idea is given human attributes. Amy Tan uses personification several times throughout her short story "Rules of The Game," which chronicles the experiences of a young chess prodigy, Waverly Place Jong. When Waverly is questioning her mother about Chinese torture as her mother roughly combs her hair, Waverly utilizes personification by saying,

"One day, as she struggled to weave a hard-toothed comb through my disobedient hair, I had a sly thought" (Tan, 1).

Waverly personifies her hair by referring to it as being disobedient. Disobedience is a human attribute and hair cannot behave a certain way. However, using personification helps describe the difficulty of combing Waverly's tough hair.
When Waverly attends her first outdoor chess tournament, she once again utilizes personification by attributing human characteristics to the light wind that begins to blow past her ears. Waverly says,

"A light wind began blowing past my ears. It whispered secrets only I could hear" (Tan, 4).

Wind is personified by having the ability to whisper into Waverly's ear like a human. Personifying the wind allows the reader to imagine Waverly's intricate thought process as she competes in her chess match.
https://literarydevices.net/personification/

Personification is a type of figurative language in which a writer gives a nonhuman subject human characteristics. In the story "Rules of the Game" (an excerpt from the novel The Joy Luck Club), Waverly, the first person narrator, often speaks figuratively in the description of her life growing up in San Francisco's Chinatown. For example, she uses personification when she describes the pigeons which she sees in the playground near the alley where she lives: "old country people sat cracking roasted watermelon seeds with their golden teeth and scattering the husks to an impatient gathering of gurgling pigeons." Impatience is usually a quality reserved for humans so by describing the pigeons as impatient as they wait for food is personification. Another example of personification occurs when Waverly is describing the chess board which her brother Vincent received as a present at the church Christmas party: "The chess board seemed to hold elaborate secrets waiting to be untangled." Again, holding secrets is definitely a human endeavor so Waverly is again personifying a nonhuman object, in this case, the chess board. 

How are nature and imagination expressed (thematically) in “Fra Lippo Lippi”?

"Fra Lippo Lippi" is a dramatic monologue in which Browning expresses his own aesthetic credo and his ideas about the subjects upon which an artist can best focus in order to achieve greatness.
The setting—Florence in the 1400s—is one in which the painter, whose actual name was Filippo Lippi, is stopped in the street at night by a gang of men who are apparently "going the rounds" and believe him to have committed some misdeed. He then proceeds to give them a brief history of his life and art. Lippi is an orphan who was taken into the care of monks. His talent for drawing has been discovered, but he comes into conflict with those around him because of the way he ultimately wishes to paint. Lippi intends his art to deal with more than spiritual matters. He is told by the monks that

Your business is not to catch men with show,
With homage to the perishable clay,
But lift them over it, ignore it all,
Make them forget there's such a thing as flesh.
Your business is to paint the souls of men—

In Browning's view, which he has Lippi express, it is precisely the natural world, which includes "the flesh," that should fire the artist's imagination. The real world—as opposed to the sterile world to which the monks wish Lippi would restrict himself—consists of

The beauty and the wonder and the power,
The shape of things, their colours, lights and shades,
Changes, surprises—and God made it all!
......yonder river's line,
The mountain round it and the sky above,
Much more the figures of man, woman, child,
These are the frame to? What's it all about?
To be passed over, despised? Or dwelt upon,
Wondered at?

The theme is that the physical world, "nature," is as much a part of the spiritual world as the rarefied and non-earthly images the religious authorities want Lippi to confine himself to.
Browning's theme in another monologue, "Andrea del Sarto," is similar. Andrea is considered the "perfect" painter, but he realizes that the artist who incorporates the imperfections of the real world in his paintings achieves a higher kind of art. The falseness of expecting perfection or completion is commented on in the lines,

Ah, but a man's reach should exceed his grasp,
Or what's a heaven for?
Similarly in the famous opening of "Rabbi Ben Ezra":
Grow old along with me!
The best is yet to be . . .

Browning is expressing the view that the struggles of life and of the aging process are to be welcomed, not rejected as signs of the "imperfection" of God's plan.
The basic view of nature and imagination in "Fra Lippo Lippi" and Browning's other works is an optimistic one. Browning sees life, including its negative and seemingly non-spiritual earthly aspects, as something to be celebrated, not despised in the tradition of the devout.

In the novel Night, discuss how Elie Wiesel is a different person from when he entered the concentration camps to when he left them.

At the beginning of the novel, Eliezer is a very religious young boy with dreams of one day learning Jewish mysticism known as Cabala. He enjoys his carefree life in Sighet where his father is a respected man, and Eliezer gets to spend his days studying the Torah and the Talmud. After the Germans occupy Hungary, Eliezer and his family are sent to various ghettos and concentration camps where they are separated from each other. In the concentration camps, Eliezer suffers malnutrition, both physical and psychological abuse, and lives in constant fear of being murdered by the Nazi soldiers. Eliezer's body endures extreme conditions as he is forced to march for miles in freezing snowstorms. The constant hunger and dire conditions force Eliezer to focus on his survival. At times, Eliezer even neglects the needs of his father in order to focus on his personal well-being. After witnessing such atrocities and surviving inhumane conditions, Eliezer is forever scarred with the traumatic images of dead bodies and grotesque living conditions. Eliezer loses his faith and does not believe that God exists. He cannot seem to understand why a benevolent God would allow such horrible acts of violence to happen. Eliezer no longer is a carefree young boy with a happy family. After losing the members of his family to Hitler's Final Solution, Eliezer is a broken individual with a negative view of humanity.

Calculus of a Single Variable, Chapter 6, 6.4, Section 6.4, Problem 64

Given equation is y' -y =y^3
An equation of the form y'+Py=Qy^n
is called as the Bernoullis equation .
so, to proceed to solve this equation we have to transform the equation into a linear equation form of first order as follows
=> y' (y^-n) +P y^(1-n)=Q
let u= y^(1-n)
=> (1-n)y^(-n)y'=u'
=> y^(-n)y' = (u')/(1-n)
so ,
y' (y^-n) +P y^(1-n)=Q
=> (u')/(1-n) +P u =Q
so this equation is now of the linear form of first order
Now,
From this equation ,
y' -y =y^3
and
y'+Py=Qy^n
on comparing we get
P=-1 , Q=1 , n=3
so the linear form of first order of the equation y' -y =y^3 is given as

=> (u')/(1-n) +P u =Q where u= y^(1-n) =y^-2
=> (u')/(1-3) +(-1) u =1
=> (-u')/2 -u=1
=> u'+2u = -2

so this linear equation is of the form
y' + py=q
p=2 , q=-2
so I.F (integrating factor ) = e^(int p dx) = e^(int 2dx) = e^(2x)

and the general solution is given as
u (I.F)=int q * (I.F) dx +c
=> u(e^(2x))= int (-2) *(e^(2x)) dx+c
=> u (e^(2x))= (-2) int (e^(2x)) dx+c
let us solve int (e^(2x)) dx
=>let t= 2x
dt = 2dx
=> int (e^(t)) dt/2
=>1/2 (int (e^(t)) dt) = 1/2 e^t = (e^(2x))/2
so, int (e^(2x)) dx =(e^(2x))/2
so ,now
u (e^(2x))= (-2) ((e^(2x))/2)+c
=>u (e^(2x))= -(e^(2x))+c
=> u = ((-(e^(2x)))+c)/(e^(2x))
but u=y^-2
so,
y^-2=((-(e^(2x)))+c)/(e^(2x))
=> y^2 = (e^(2x))/((-(e^(2x)))+c)
=> y = sqrt((e^(2x))/((-(e^(2x)))+c))

=> y = e^x/(sqrt(c-e^(2x)))
the general solution.