Advertisements
Advertisements
Question
Show that the general solution of differential equation `"dy"/"dx" + ("y"^2 + "y" + 1)/("x"^2 + "x" + 1) = 0` is given by (x + y + 1) = (1 - x - y - 2xy).
Solution
`"dy"/"dx" + ("y"^2 + "y" + 1)/("x"^2 + "x" + 1) = 0`
∴ `"dy"/"dx" = - (("y"^2 + "y" + 1)/("x"^2 + "x" + 1))`
∴ `1/("y"^2 + "y" + 1)"dy" = - 1/("x"^2 + "x" + 1)"dx"`
Integrating both sides, we get
`int 1/("y"^2 + "y" + 1)"dy" = - int 1/("x"^2 + "x" + 1)"dx"`
∴ `int 1/(("y"^2 + "y" + 1/4) + 3/4) "dy" = - int 1/(("x"^2 + "x" + 1/4) + 3/4)"dx"`
∴ `int 1/(("y" + 1/2)^2 + (sqrt3/2)^2)"dy" = - int 1/(("x" + 1/2)^2 + (sqrt3/2)^2)"dx"`
∴ `1/(sqrt3/2) tan^-1 [("y" + 1/2)/(sqrt3/2)] = - 1/((sqrt3/2)) tan^-1 [("x" + 1/2)/(sqrt3/2)] + "c"_1`
∴ `2/sqrt3 tan^-1 (("2y" + 1)/sqrt3) + 2/sqrt3 tan^-1 (("2x" + 1)/sqrt3) = "c"_1`
∴ `2/sqrt3 tan^-1 [((("2y" + 1)/sqrt3) + (("2x" + 1)/sqrt3))/(1 - (("2y" + 3)/sqrt3)(("2x" + 1)/sqrt3))] = "c"_1`
∴ `tan^-1 ((("2y" + 1 + 2"x" + 1)/sqrt3))/(((3 - 4"xy" - 2"y" - 2"x" - 1)/3)) = sqrt3/2 "c"_1`
∴ `(("2y" + "2x" + 2)sqrt3)/(2 - "2x" - 2"y" - 4"xy") = tan (sqrt3/2 "c"_1)`
∴ `("x + y + z")/(1 - "x" - "y" - 2"xy") = 1/sqrt3 tan (sqrt3/2 "c"_1) = "c"`, where c = `1/sqrt3 tan (sqrt3/2 "c"_1)`
∴ (x + y + 1) = c(1 - x - y - 2xy)
This is the general solution.
APPEARS IN
RELATED QUESTIONS
If a body cools from 80°C to 50°C at room temperature of 25°C in 30 minutes, find the temperature of the body after 1 hour.
The rate of decay of certain substances is directly proportional to the amount present at that instant. Initially, there is 25 gm of certain substance and two hours later it is found that 9 gm are left. Find the amount left after one more hour.
Find the population of a city at any time t, given that the rate of increase of population is proportional to the population at that instant and that in a period of 40 years, the population increased from 30,000 to 40,000.
A right circular cone has height 9 cm and radius of the base 5 cm. It is inverted and water is poured into it. If at any instant the water level rises at the rate of `(pi/"A")`cm/sec, where A is the area of the water surface A at that instant, show that the vessel will be full in 75 seconds.
Assume that a spherical raindrop evaporates at a rate proportional to its surface area. If its radius originally is 3 mm and 1 hour later has been reduced to 2 mm, find an expression for the radius of the raindrop at any time t.
The rate of growth of the population of a city at any time t is proportional to the size of the population. For a certain city, it is found that the constant of proportionality is 0.04. Find the population of the city after 25 years, if the initial population is 10,000. [Take e = 2.7182]
Choose the correct option from the given alternatives:
If the surrounding air is kept at 20° C and a body cools from 80° C to 70° C in 5 minutes, the temperature of the body after 15 minutes will be
A person’s assets start reducing in such a way that the rate of reduction of assets is proportional to the square root of the assets existing at that moment. If the assets at the beginning ax ‘ 10 lakhs and they dwindle down to ‘ 10,000 after 2 years, show that the person will be bankrupt in `2 2/9` years from the start.
The population of a town increases at a rate proportional to the population at that time. If the population increases from 40 thousands to 60 thousands in 40 years, what will be the population in another 20 years?
(Given: `sqrt(3/2)= 1.2247)`
The rate of growth of population is proportional to the number present. If the population doubled in the last 25 years and the present population is 1 lac, when will the city have population 4,00,000?
The rate of growth of bacteria is proportional to the number present. If initially, there were 1000 bacteria and the number doubles in 1 hour, find the number of bacteria after `5/2` hours `("Given" sqrt(2) = 1.414)`
Choose the correct alternative:
Bacterial increases at the rate proportional to the number present. If original number M doubles in 3 hours, then number of bacteria will be 4M in
Choose the correct alternative:
The integrating factor of `("d"y)/("d"x) + y` = e–x is
Choose the correct alternative:
The integrating factor of `("d"^2y)/("d"x^2) - y` = ex, is e–x, then its solution is
Choose the correct alternative:
The solution of `dy/dx` = 1 is ______.
Choose the correct alternative:
The solution of `("d"y)/("d"x) + x^2/y^2` = 0 is
The integrating factor of the differential equation `("d"y)/("d"x) - y` = x is ______
Bacteria increases at the rate proportional to the number of bacteria present. If the original number N doubles in 4 hours, find in how many hours the number of bacteria will be 16N.
Solution: Let x be the number of bacteria in the culture at time t.
Then the rate of increase of x is `("d"x)/"dt"` which is proportional to x.
∴ `("d"x)/"dt" ∝ x`
∴ `("d"x)/"dt"` = kx, where k is a constant
∴ `("d"x)/x` = kdt
On integrating, we get
`int ("d"x)/x = "k" int "dt"`
∴ log x = kt + c .....(1)
∴ x = aekt where a = ec
Initially, i.e.,when t = 0, let x = N
∴ N = aek(0)
∴ a = `square`
∴ a = N, x = Nekt ......(2)
When t = 4, x = 2N
From equation (2), 2N = Ne4k
∴ e4k = 2
∴ ek = `square`
Now we have to find out t, when x = 16N
From equation (2),
16N = Nekt
∴ 16 = ekt
∴ `"t"/4 = square` hours
Hence, number of bacteria will be 16N in `square` hours
If the population grows at the rate of 8% per year, then the time taken for the population to be doubled, is (Given log 2 = 0.6912).
The rate of decay of certain substance is directly proportional to the amount present at that instant. Initially, there are 27 gm of certain substance and 3 h later it is found that 8 gm are left, then the amount left after one more hour is ______.
The bacteria increases at the rate proportional to the number of bacteria present. If the original number 'N' doubles in 4 h, then the number of bacteria in 12 h will be ____________.
If the lengths of the transverse axis and the latus rectum of a hyperbola are 6 and `8/3` respectively, then the equation of the hyperbola is ______.
The population of a town increases at a rate proportional to the population at that time. If the population increases from 26,000 to 39,000 in 50 years, then the population in another 25 years will be ______ `(sqrt(3/2) = 1.225)`
The length of the perimeter of a sector of a circle is 24 cm, the maximum area of the sector is ______.
If a curve y = f(x) passes through the point (1, - 1) and satisfies the differential equation, y (1 + xy) dx = x dy, then `f(-1/2)` is equal to ______
The rate of disintegration of a radioactive element at time t is proportional to its mass at that time. The original mass of 800 gm will disintegrate into its mass of 400 gm after 5 days. Find the mass remaining after 30 days.
Solution: If x is the amount of material present at time t then `dx/dt = square`, where k is constant of proportionality.
`int dx/x = square + c`
∴ logx = `square`
x = `square` = `square`.ec
∴ x = `square`.a where a = ec
At t = 0, x = 800
∴ a = `square`
At t = 5, x = 400
∴ e–5k = `square`
Now when t = 30
x = `square` × `square` = 800 × (e–5k)6 = 800 × `square` = `square`.
The mass remaining after 30 days will be `square` mg.
If `(dy)/(dx)` = y + 3 > 0 and y = (0) = 2, then y (in 2) is equal to ______.
Bacteria increase at the rate proportional to the number of bacteria present. If the original number N doubles in 3 hours, find in how many hours the number of bacteria will be 4N?
