Calculus of a Single Variable, Chapter 6, 6.4, Section 6.4, Problem 66

Given equation is yy'-2y^2=e^x
=> y' -2y=e^x y^(-1)
An equation of the form y'+Py=Qy^n
is called the Bernoulli 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' -2y=e^x y^(-1)
and
y'+Py=Qy^n
on comparing we get
P=-2 Q=e^x , n=-1
so the linear form of first order of the equation y' -2y=e^x y^(-1) is given as

=> (u')/(1-n) +P u =Q where u= y^(1-n) =y^2
=> (u')/(1-(-1)) +(-2)u =e^x
=> (u')/2 -2u=e^x
=> (u')-4u = 2e^x

so this linear equation is of the form
u' + pu=q
p=-4 , q=2e^x
so I.F (integrating factor ) = e^(int p dx) = e^(int -4dx) = e^(-4x)

and the general solution is given as
u (I.F)=int q * (I.F) dx +c
=> u(e^(-4x))= int (2e^x) *(e^(-4x)) dx+c
=> u(e^(-4x))= int (2e^x) *(e^(-4x)) dx+c
=> u(e^(-4x))= 2 int (e^(-3x)) dx+c
=>u(e^(-4x))= 2 int (e^(-3x)) dx+c
=>u(e^(-4x))= 2 (1/(-3)*e^(-3x))+c as int e^(ax) dx = 1/a e^(ax).
=>u(e^(-4x))= (-2/3)*e^(-3x)+c
=> u = ((-2/3)*e^(-3x)+c)/(e^(-4x))
but u= y^2 so ,
y^2 = ((-2/3)*e^(-3x)+c)/(e^(-4x))
y= sqrt((-2/3e^(-3x)+c)/(e^(-4x)))
=sqrt((-2/3e^(-3x)+c)*(e^(4x)))
= sqrt((-2/3e^(x)+ce^(4x)))
=e^(x/2)sqrt((-2+3ce^(3x))/3)
is the general solution.

Comments

Post a Comment

Popular posts from this blog

How does Bilbo show leadership and courage in The Hobbit?

Bilbo’s increase in confidence and courage is the defining aspect of his character arc. At the beginning of the novel, he is shut away content in his Hobbit hole, happy to remain separate and coddled from the rest of the world. In the movie, Bilbo chooses to join the dwarves on their quest out of a sense for adventure. In the book, however, Bilbo wants nothing to do with them and only joins because he had been duped into signing a contract. Despite being on the adventure somewhat against his will, Bilbo still proves himself time and again, saving the dwarves from the spiders, the trolls, and a number of other dangers they encounter along the way. Bilbo’s character arc has even been cited as a sort of Christian metaphor for the growth of the soul, though how much Tolkien intended on this front is up for debate. From a thematic standpoint, the ring of invisibility also plays into Bilbo’s brand of bravery. If Hobbits are the overlooked creatures of Middle Earth, then it is noteworthy that...
Read more

What was the effect of World War II on African Americans?

According to historians John Hope Franklin and Alfred A. Moss, Jr., more than three million black men registered for the Selective Service. However, they were refused at a rate of 18.2 percent, compared to just 8.5 percent for whites.  Discrimination was rampant in the armed forces, though black soldiers had more opportunities than in the First World War. They were included in the infantry, the coast and field artillery, the engineer corps, the medical corps, and many other branches. When the Women's Army Corps was organized (WAC), black women were included. By the end of the war, more than 4,000 women had enlisted in the organization. The next step was to get black soldiers recommended for advanced training at officer training schools. Blacks were not admitted to officer training schools in high numbers until the Secretary of War issued an order stating that blacks were to be admitted. The integration of the military began in 1945, first on the war front in Germany. In 1948, sever...
Read more

How is Jack in William Golding's The Lord of the Flies presented as militaristic?

Jack, who has red hair that symbolizes his association with blood, constantly carries a knife around with him. He is responsible for hunting, and when he at first fails to kill a pig, "he snatched his knife out of the sheath and slammed it into a tree trunk" (page 31). As the novel goes on, Jack becomes increasingly more violent, and his knife is often withdrawn from his sheath. He relies on violence to increase his power. Jack resents Piggy's intelligence, and he wants to rid Piggy of his glasses, the symbol of Piggy's learning. For example, when the boys are lighting a fire in the hope of being rescued, Jack says, "His specs--use them as burning glasses!" (page 40). Jack prefers to use Piggy's glasses to make a fire than to allow Piggy, the resident intellectual, to see. Later, Jack, constantly seen with a spear in hand and often bloodied from hunting, gives up all attempts to be rescued. Instead, he uses his violence, symbolized by the spear, to take ...
Read more