Advertisements
Advertisements
Solve the following differential equation
`yx ("d"y)/("d"x)` = x^{2} + 2y^{2}
Advertisements
Solution
`yx ("d"y)/("d"x)` = x^{2} + 2y^{2}
∴ `("d"y)/("d"x) = (x^2 + 2y^2)/(xy)` ......(i)
Put y = vx ......(ii)
Differentiating w.r.t. x, we get
`("d"y)/("d"x) = "v" + x "dv"/("d"x)` ......(iii)
Substituting (ii) and (iii) in (i), we get
`"v" + x "dv"/("d"x) = (x^2 + 2"v"^2x^2)/(x("v"x))`
∴ `"v" + x "dv"/("d"x) = (x^2(1 + 2"v"^2))/(x^2"v")`
∴ `x "dv"/("d"x) = (1 + 2"v"^2)/"v"  "v"`
= `(1 + "v"^2)/"v"`
∴ `"v"/(1 + "v"^2) "dv" = 1/x "d"x`
Integrating on both sides, we get
`1/ int (2"v")/(1 +"v"^2) "dv" = int "dv"/x`
∴ `1/2 log1 + "v"^2` = log x + log c
∴ log 1 + c^{2} = 2 og x + 2log c
= log x^{2} + log c^{2}
∴ log 1 + v^{2} = log c^{2}x^{2}
∴ 1 + v^{2} = c^{2}x^{2}
∴ `1 + y^2/x^2` = c^{2}x^{2}
∴ x^{2} + y^{2} = c^{2}x^{4 }
RELATED QUESTIONS
Prove that :
`int_0^(2a)f(x)dx=int_0^af(x)dx+int_0^af(2ax)dx`
If 1, `omega` and `omega^2` are the cube roots of unity, prove `(a + b omega + c omega^2)/(c + s omega + b omega^2) = omega^2`
Solve the equation for x: `sin^(1) 5/x + sin^(1) 12/x = pi/2, x != 0`
Assume that a rain drop evaporates at a rate proportional to its surface area. Form a differential equation involving the rate of change of the radius of the rain drop.
Verify that y = cx + 2c^{2} is a solution of the differential equation
Verify that y^{2} = 4a (x + a) is a solution of the differential equations
\[y\left\{ 1  \left( \frac{dy}{dx} \right)^2 \right\} = 2x\frac{dy}{dx}\]
For the following differential equation verify that the accompanying function is a solution:
Differential equation  Function 
\[y = \left( \frac{dy}{dx} \right)^2\]

\[y = \frac{1}{4} \left( x \pm a \right)^2\]

(sin x + cos x) dy + (cos x − sin x) dx = 0
C' (x) = 2 + 0.15 x ; C(0) = 100
(1 + x^{2}) dy = xy dx
x cos y dy = (xe^{x} log x + e^{x}) dx
(e^{y} + 1) cos x dx + e^{y} sin x dy = 0
tan y dx + sec^{2} y tan x dy = 0
(y^{2} + 1) dx − (x^{2} + 1) dy = 0
Solve the differential equation \[\frac{dy}{dx} = \frac{2x\left( \log x + 1 \right)}{\sin y + y \cos y}\], given that y = 0, when x = 1.
In a bank principal increases at the rate of 5% per year. An amount of Rs 1000 is deposited with this bank, how much will it worth after 10 years (e^{0.5} = 1.648).
x^{2} dy + y (x + y) dx = 0
3x^{2} dy = (3xy + y^{2}) dx
Solve the following initial value problem:
\[y' + y = e^x , y\left( 0 \right) = \frac{1}{2}\]
Solve the following initial value problem:
\[x\frac{dy}{dx} + y = x \cos x + \sin x, y\left( \frac{\pi}{2} \right) = 1\]
The surface area of a balloon being inflated, changes at a rate proportional to time t. If initially its radius is 1 unit and after 3 seconds it is 2 units, find the radius after time t.
The rate of growth of a population is proportional to the number present. If the population of a city doubled in the past 25 years, and the present population is 100000, when will the city have a population of 500000?
In a simple circuit of resistance R, self inductance L and voltage E, the current `i` at any time `t` is given by L \[\frac{di}{dt}\]+ R i = E. If E is constant and initially no current passes through the circuit, prove that \[i = \frac{E}{R}\left\{ 1  e^{ \left( R/L \right)t} \right\}.\]
Find the equation of the curve passing through the point \[\left( 1, \frac{\pi}{4} \right)\] and tangent at any point of which makes an angle tan^{−1 }\[\left( \frac{y}{x}  \cos^2 \frac{y}{x} \right)\] with xaxis.
The rate of increase of bacteria in a culture is proportional to the number of bacteria present and it is found that the number doubles in 6 hours. Prove that the bacteria becomes 8 times at the end of 18 hours.
Find the equation of the curve passing through the point (0, 1) if the slope of the tangent to the curve at each of its point is equal to the sum of the abscissa and the product of the abscissa and the ordinate of the point.
Define a differential equation.
Write the differential equation representing the family of straight lines y = Cx + 5, where C is an arbitrary constant.
Integrating factor of the differential equation cos \[x\frac{dy}{dx} + y\] sin x = 1, is
Which of the following transformations reduce the differential equation \[\frac{dz}{dx} + \frac{z}{x}\log z = \frac{z}{x^2} \left( \log z \right)^2\] into the form \[\frac{du}{dx} + P\left( x \right) u = Q\left( x \right)\]
If x^{m}y^{n} = (x + y)^{m+n}, prove that \[\frac{dy}{dx} = \frac{y}{x} .\]
y^{2} dx + (x^{2} − xy + y^{2}) dy = 0
Form the differential equation of the family of parabolas having vertex at origin and axis along positive yaxis.
If a + ib = `("x" + "iy")/("x"  "iy"),` prove that `"a"^2 +"b"^2 = 1` and `"b"/"a" = (2"xy")/("x"^2  "y"^2)`
Find the coordinates of the centre, foci and equation of directrix of the hyperbola x^{2} – 3y^{2} – 4x = 8.
Solve the differential equation:
`"x"("dy")/("dx")+"y"=3"x"^22`
Find the equation of the plane passing through the point (1, 2, 1) and perpendicular to the line joining the points A(3, 2, 1) and B(1, 4, 2).
The price of six different commodities for years 2009 and year 2011 are as follows:
Commodities  A  B  C  D  E  F 
Price in 2009 (₹) 
35  80  25  30  80  x 
Price in 2011 (₹)  50  y  45  70  120  105 
The Index number for the year 2011 taking 2009 as the base year for the above data was calculated to be 125. Find the values of x andy if the total price in 2009 is ₹ 360.
Choose the correct option from the given alternatives:
The differential equation `"y" "dy"/"dx" + "x" = 0` represents family of
Choose the correct option from the given alternatives:
The solution of `1/"x" * "dy"/"dx" = tan^1 "x"` is
In each of the following examples, verify that the given function is a solution of the corresponding differential equation.
Solution  D.E. 
xy = log y +k  y' (1xy) =y2 
In each of the following examples, verify that the given function is a solution of the corresponding differential equation.
Solution  D.E. 
y = e^{x}  `dy/ dx= y` 
Determine the order and degree of the following differential equations.
Solution  D.E. 
y = 1 − logx  `x^2(d^2y)/dx^2 = 1` 
Find the differential equation whose general solution is
x^{3} + y^{3} = 35ax.
Form the differential equation from the relation x^{2 }+ 4y^{2 }= 4b^{2}
For each of the following differential equations find the particular solution.
`(x + 1) dy/dx − 1 = 2e^(−y)` ,
when y = 0, x = 1
Solve the following differential equation.
xdx + 2y dy = 0
Solve the following differential equation.
`xy dy/dx = x^2 + 2y^2`
Solve the following differential equation.
`dy/dx + y = e ^x`
Solve the following differential equation.
`dy/dx + y` = 3
Solve the following differential equation.
y dx + (x  y^{2} ) dy = 0
Solve the following differential equation.
dr + (2r)dθ= 8dθ
Choose the correct alternative.
The differential equation of `y = k_1e^x+ k_2 e^x` is ______.
The solution of `dy/dx + x^2/y^2 = 0` is ______
A solution of a differential equation which can be obtained from the general solution by giving particular values to the arbitrary constants is called ___________ solution.
Select and write the correct alternative from the given option for the question
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
Select and write the correct alternative from the given option for the question
The differential equation of y = Ae^{5x} + Be^{–5x} is
Select and write the correct alternative from the given option for the question
Differential equation of the function c + 4yx = 0 is
Solve the differential equation sec^{2}y tan x dy + sec^{2}x tan y dx = 0
Solve the differential equation `("d"y)/("d"x) + y` = e^{−x}
Solve `("d"y)/("d"x) = (x + y + 1)/(x + y  1)` when x = `2/3`, y = `1/3`
Solve the differential equation xdx + 2ydy = 0
Solve the differential equation (x^{2} – yx^{2})dy + (y^{2} + xy^{2})dx = 0
Solve the following differential equation `("d"y)/("d"x)` = x^{2}y + y
Solve: `("d"y)/("d"x) + 2/xy` = x^{2}
For the differential equation, find the particular solution (x – y^{2}x) dx – (y + x^{2}y) dy = 0 when x = 2, y = 0
Solve the following differential equation y log y = `(log y  x) ("d"y)/("d"x)`
For the differential equation, find the particular solution
`("d"y)/("d"x)` = (4x +y + 1), when y = 1, x = 0
Solve the following differential equation y^{2}dx + (xy + x^{2}) dy = 0
Solve the following differential equation
`x^2 ("d"y)/("d"x)` = x^{2} + xy − y^{2}
Choose the correct alternative:
Solution of the equation `x("d"y)/("d"x)` = y log y is
Choose the correct alternative:
General solution of `y  x ("d"y)/("d"x)` = 0 is
The function y = cx is the solution of differential equation `("d"y)/("d"x) = y/x`
Solve the following differential equation `("d"y)/("d"x)` = x^{2}y + y
Verify y = log x + c is the solution of differential equation `x ("d"^2y)/("d"x^2) + ("d"y)/("d"x)` = 0
Solve the following differential equation
`y log y ("d"x)/("d"y) + x` = log y
Verify y = `a + b/x` is solution of `x(d^2y)/(dx^2) + 2 (dy)/(dx)` = 0
y = `a + b/x`
`(dy)/(dx) = square`
`(d^2y)/(dx^2) = square`
Consider `x(d^2y)/(dx^2) + 2(dy)/(dx)`
= `x square + 2 square`
= `square`
Hence y = `a + b/x` is solution of `square`
Solve the following differential equation `("d"y)/("d"x)` = cos(x + y)
Solution: `("d"y)/("d"x)` = cos(x + y) ......(1)
Put `square`
∴ `1 + ("d"y)/("d"x) = "dv"/("d"x)`
∴ `("d"y)/("d"x) = "dv"/("d"x)  1`
∴ (1) becomes `"dv"/("d"x)  1` = cos v
∴ `"dv"/("d"x)` = 1 + cos v
∴ `square` dv = dx
Integrating, we get
`int 1/(1 + cos "v") "d"v = int "d"x`
∴ `int 1/(2cos^2 ("v"/2)) "dv" = int "d"x`
∴ `1/2 int square "dv" = int "d"x`
∴ `1/2* (tan("v"/2))/(1/2)` = x + c
∴ `square` = x + c
Solve the differential equation `"dy"/"dx"` = 1 + x + y^{2} + xy^{2}, when y = 0, x = 0.
Solution of `x("d"y)/("d"x) = y + x tan y/x` is `sin(y/x)` = cx
lf the straight lines `ax + by + p` = 0 and `x cos alpha + y sin alpha = p` are inclined at an angle π/4 and concurrent with the straight line `x sin alpha  y cos alpha` = 0, then the value of `a^2 + b^2` is
There are n students in a school. If r % among the students are 12 years or younger, which of the following expressions represents the number of students who are older than 12?
If `y = log_2 log_2(x)` then `(dy)/(dx)` =
A man is moving away from a tower 41.6 m high at a rate of 2 m/s. If the eye level of the man is 1.6 m above the ground, then the rate at which the angle of elevation of the top of the tower changes, when he is at a distance of 30 m from the foot of the tower, is
`d/(dx)(tan^1 (sqrt(1 + x^2)  1)/x)` is equal to:
The differential equation (1 + y^{2})x dx – (1 + x^{2})y dy = 0 represents a family of:
Solve the differential equation
`y (dy)/(dx) + x` = 0
Solve the differential equation
`x + y dy/dx` = x^{2} + y^{2}
Solve the differential equation `dy/dx + xy = xy^2` and find the particular solution when y = 4, x = 1.