Calculus of a Single Variable, Chapter 6, 6.4, Section 6.4, Problem 23

Given xdy = (x + y + 2)dx
=> xdy/dx = (x + y + 2)
=> y'=1+y/x+2/x
=> y'-y/x = 1+2/x
=> y'-y/x =(x+2)/x
when the first order linear ordinary differential equation has the form of
y'+p(x)y=q(x)
then the general solution is ,
y(x)=((int e^(int p(x) dx) *q(x)) dx +c)/e^(int p(x) dx)
so,
y'-y/x =(x+2)/x--------(1)
y'+p(x)y=q(x)---------(2)
on comparing both we get,
p(x) = -1/x and q(x)=(x+2)/x
so on solving with the above general solution we get:
y(x)=((int e^(int p(x) dx) *q(x)) dx +c)/e^(int p(x) dx)
=((int e^(int (-1/x) dx) *((x+2)/x)) dx +c)/e^(int (-1/x) dx)
first we shall solve
e^(int (-1/x) dx)=e^(ln(1/x)) = (1/x)
so
proceeding further, we get
y(x) =((int e^(int (-1/x) dx) *((x+2)/x)) dx +c)/e^(int (-1/x) dx)
=((int (1/x) *((x+2)/x)) dx +c)/(1/x)
=(int (1/x) *((1+(2)/x)) dx +c)/(1/x)
= x(ln(x)-2/x +c)
so now let us find the particular solution of differential equation at y(1)=10
y(1) = 1(ln(1)-2/1 +c)
=> 10 = 0-2+c
c=12
y(x) =x(ln(x)-2/x +12) is the solution

Explain title of the novel White Teeth. Where does Smith explicitly reference the title in the novel and what does the scene demonstrate?

The title of White Teeth is instructive because it encapsulates one of the book's main themes—how society reinforces the harmful notion that our shared humanity is challenged and undermined by differences of race and culture. However, all of us, whatever our skin color, nationality, or racial heritage, have the same white teeth. It is the color of our teeth that identifies us all as being part of the same human family.
Paradoxically, white teeth can also symbolize racial differences. The children discover this in chapter 7 when they bring food from their school's Harvest Festival to an old man by the name of Mr. Hamilton. The old war veteran is an unabashed racist, and tells the children about the time he served with the British Army in the Congo. At that time, white teeth served to accentuate racial distinctions:

Clean white teeth are not always wise, now are they? Par exemplum: when I was in the Congo, the only way I could identify the n****r was by the whiteness of his teeth, if you see what I mean. Horrid business. Dark as buggery, it was. And they died because of it, you see? Poor bastards (p.113).

Ironically, Mr. Hamilton only has false teeth. This can be interpreted as further evidence of his general lack of humanity.

Calculus of a Single Variable, Chapter 6, 6.4, Section 6.4, Problem 62

Bernoulli equation has form y'+P(x)y=Q(x)y^alpha. We can convert any such equation into linear equation differential equation by using substitution z=y^(1-alpha). To learn more about this method of solving Bernoulli equation check out the links below.
We will show a somewhat different method which can also be used for solving linear equations.
y'+y/x=x sqrt(y)
Make substitution y=uv => y'=u'v+uv'
u'v+uv'+(uv)/x=x sqrt(uv)
v(u'+u/x)+uv'=x sqrt(uv) (1)
Let us now assume that the expression in brackets equals zero (we can choose almost anything instead of zero, but this makes the calculation easier) in order to calculate u.
u'+u/x=0
u'=-u/x
(du)/u=-dx/x
Integrating both sides yields
ln u=-ln x
ln u=ln x^-1
u=x^-1
Plugging that into (1) gives
x^-1 v'=x sqrt(x^-1 v)
Now we multiply by x to get only v' on the left hand side.
v'=x^2x^(-1/2)sqrt v
v'=x^(3/2)sqrt v
(dv)/sqrt v=x^(3/2)dx
Integrating the above equation gives us
2sqrt v=(2x^(5/2)+C)/5
Divide by 2 and square the whole equation in order to get v.
v=(x^5+2Cx^(5/2)+C^2)/25
Now we just plug the obtained u and v into the substitution to get the final result.
y=x^-1 cdot (x^5+2Cx^(5/2)+C^2)/25
y=(x^5+2Cx^(5/2)+C^2)/(25x) lArr The general solution.

How does she explain where she is going to the hunter? How do we know how much old Phoenix loves her grandson?

In the story, Phoenix Jackson meets a young white hunter on her way to town. When the hunter asks Phoenix where she is going, she tells him that she is going to town. Upon hearing her answer, the hunter exclaims that the way to town is too far for an old woman like her. He then advises her to head for home.
For her part, Phoenix doesn't reveal exactly why she is going to town. She merely satisfies his curiosity with a general answer: "The time come around." From her answer, the hunter speculates that she is going to town to see Santa Claus. He assumes that she must be like other "old colored people" who make the journey to town for this purpose. Essentially, Phoenix Jackson provides no concrete details about her reason for going to town. She merely explains that she goes to town because she must.
We know how much Phoenix Jackson loves her grandson by her willingness to endure a long, arduous journey for his sake. Additionally, old Phoenix makes the journey regularly. During each trip, she must brave varied dangers. Her exchange with the white hunter reveals the low status of African Americans in her society.
When the hunter points his gun at Phoenix and asks her whether she is afraid of it, she remains calm; her reply is a simple "No, sir, I seen plenty go off closer by, in my day, and for less than what I done." Phoenix knows that her safety is not promised in such a society; yet, she makes the journey because she loves her grandson.

How is Rikki-tikki-tavi courageous?

The defining character traits of a mongoose are their inherent curiosity and bravery. Rikki-tikki-tavi’s character does not deviate from the norm, and he displays his bravery numerous times in the short story. After surviving a turbulent flood, he ends up in a lovely bungalow, which is inhabited by a pleasant British family. Rikki-tikki initially demonstrates his bravery by standing face-to-face with Nag in the garden and killing a deadly sand snake named Karait. Rikki-tikki once again displays his bravery by fighting and killing Nag in the family’s bathroom to prevent him from harming Teddy’s father. Despite being much smaller, Rikki-tikki defeats the large, intimidating male cobra. In Rikki-tikki’s most impressive display of bravery, he saves Teddy’s life and defeats Nagaina in her underground nest. Entering the cobra’s nest is an extremely courageous action and Rikki-tikki risks his life during his deadly fight with Nagaina. Overall, Rikki-tikki demonstrates his bravery by defeating the deadly snakes living on the compound and protecting the British family from the threatening cobras.

Bravery is one of Rikki-tikki-tavi's defining characteristics, and he displays great courage at many points throughout the story. He takes very seriously his role as protector and defender of Teddy and his family, and Rikki-tikki doesn't hold back on confronting any threatening snakes, no matter how poisonous they are. Nag and Nagaina are two fearsome cobras; they regard the family's garden as belonging to them as they were there before the bungalow was built. Not surprisingly, they aren't too happy to see human beings and a mongoose occupying their territory. But Rikki-tikki doesn't back down and confronts the dangerous cobras, killing Nagainia after Teddy's father shoots Nag.
In an even greater act of bravery, he also challenges Karait, a small, venomous sand snake even more deadly than a cobra. In fact, he even kills the snake, saving Teddy from certain death in the process.

How does Clarisse feel about technology in Fahrenheit 451?

In Fahrenheit 451, Clarisse McClellan is utterly disinterested in technology. She explains this to Montag when they first meet:

"I rarely watch the parlour walls or go to races or Fun Parks."

For Clarisse, life is about smelling the rain and collecting butterflies, not about spending time driving too fast or watching the family. Even at school, she finds that activities like "TV class" and "transcription history" are utterly pointless since they do not encourage students to think or ask questions:

"They run us so ragged by the end of the day we can't do anything but go to bed or head for a Fun Park to bully people around."

Because she is so disinterested in the technology which is available to her, she has been cast out by her society: she is under surveillance, for example, and is forced to see a psychiatrist. Labelled as "anti-social," Clarisse provides the impetus for Montag's rebellion but tragically loses her life, probably as a result of a hit-and-run accident.

What does Marta represent?

In "The Falling Girl," Marta is a young nineteen-year-old girl that represents a progression through life, and her fall represents expectations of society. As she falls, she rises in society until she reaches the lower levels. Marta begins the story as a young girl and ends as an older lady. This part of the story seems conventional. However, consider the time of day in which the story begins and ends. Dawn would normally be associated with a beginning and dusk with an end. This is not the case in "The Falling Girl." The author flips some of our expectations as readers. For example, Marta is nineteen when she begins her fall, and she sees "the city below shining in the dusk." Marta ages as she falls, and as she nears the street level, it is early in the morning.
The author shares that "flights of that kind" are not a rare occurrence in the skyscraper. People pay more to live in the upper floors so that they can see the beautiful girls fall. Near the bottom, the tenants are denied the privilege of the youthful falling girls, but they do get to "hear the thud when they touch the ground." This is a reflection on society. As Marta falls, she notices that girls dressed more beautifully are falling faster. The author refers to a "contest." Marta feels that they will reach the target before she does. This represents the race for the top in society. However, in Marta's case, it is a race for the bottom as she falls.