Advertisements
Advertisements
Choose the correct alternative:
Differential equation of the function c + 4yx = 0 is
Options
`xy + ("d"y)/("d"x)` = 0
`x ("d"y)/("d"x) + y` = 0
`("d"y)/("d"x)  4xy` =0
`x ("d"y)/("d"x) + 1` = 0
Advertisements
Solution
`x ("d"y)/("d"x) + y` = 0
APPEARS IN
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 \[y = ce^{tan^{ 1}} x\] is a solution of the differential equation \[\left( 1 + x^2 \right)\frac{d^2 y}{d x^2} + \left( 2x  1 \right)\frac{dy}{dx} = 0\]
For the following differential equation verify that the accompanying function is a solution:
Differential equation  Function 
\[x\frac{dy}{dx} = y\]

y = ax 
For the following differential equation verify that the accompanying function is a solution:
Differential equation  Function 
\[x + y\frac{dy}{dx} = 0\]

\[y = \pm \sqrt{a^2  x^2}\]

x cos y dy = (xe^{x} log x + e^{x}) dx
(e^{y} + 1) cos x dx + e^{y} sin x dy = 0
(y + xy) dx + (x − xy^{2}) dy = 0
Solve the following differential equation:
\[\left( 1 + y^2 \right) \tan^{ 1} xdx + 2y\left( 1 + x^2 \right)dy = 0\]
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} − y^{2}) dx − 2xy dy = 0
Solve the following initial value problem:
\[y' + y = e^x , y\left( 0 \right) = \frac{1}{2}\]
Solve the following initial value problem:
\[x\frac{dy}{dx}  y = \log x, y\left( 1 \right) = 0\]
Solve the following initial value problem:
\[\tan x\left( \frac{dy}{dx} \right) = 2x\tan x + x^2  y; \tan x \neq 0\] given that y = 0 when \[x = \frac{\pi}{2}\]
In a culture, the bacteria count is 100000. The number is increased by 10% in 2 hours. In how many hours will the count reach 200000, if the rate of growth of bacteria is proportional to the number present?
The population of a city increases at a rate proportional to the number of inhabitants present at any time t. If the population of the city was 200000 in 1990 and 250000 in 2000, what will be the population in 2010?
If the marginal cost of manufacturing a certain item is given by C' (x) = \[\frac{dC}{dx}\] = 2 + 0.15 x. Find the total cost function C (x), given that C (0) = 100.
The decay rate of radium at any time t is proportional to its mass at that time. Find the time when the mass will be halved of its initial mass.
The slope of the tangent at a point P (x, y) on a curve is \[\frac{ x}{y}\]. If the curve passes through the point (3, −4), find the equation of the curve.
Find the curve for which the intercept cutoff by a tangent on xaxis is equal to four times the ordinate of the point of contact.
The tangent at any point (x, y) of a curve makes an angle tan^{−1}(2x + 3y) with xaxis. Find the equation of the curve if it passes through (1, 2).
Find the equation of the curve which passes through the point (3, −4) and has the slope \[\frac{2y}{x}\] at any point (x, y) on it.
Find the equation of the curve which passes through the origin and has the slope x + 3y− 1 at any point (x, y) on it.
The slope of the tangent at each point of a curve is equal to the sum of the coordinates of the point. Find the curve that passes through the origin.
Find the equation of the curve that passes through the point (0, a) and is such that at any point (x, y) on it, the product of its slope and the ordinate is equal to the abscissa.
The integrating factor of the differential equation (x log x)
\[\frac{dy}{dx} + y = 2 \log x\], is given by
The equation of the curve whose slope is given by \[\frac{dy}{dx} = \frac{2y}{x}; x > 0, y > 0\] and which passes through the point (1, 1) is
What is integrating factor of \[\frac{dy}{dx}\] + y sec x = tan x?
Integrating factor of the differential equation cos \[x\frac{dy}{dx} + y \sin x = 1\], is
If x^{m}y^{n} = (x + y)^{m+n}, prove that \[\frac{dy}{dx} = \frac{y}{x} .\]
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
For each of the following differential equations find the particular solution.
`(x + 1) dy/dx − 1 = 2e^(−y)` ,
when y = 0, x = 1
For each of the following differential equations find the particular solution.
`y (1 + logx)dx/dy  x log x = 0`,
when x=e, y = e^{2}.
Solve the following differential equation.
xdx + 2y dy = 0
Solve the following differential equation.
y^{2} dx + (xy + x^{2} ) dy = 0
Solve the following differential equation.
`dy /dx +(x2 y)/ (2x y)= 0`
Solve the following differential equation.
`(x + y) dy/dx = 1`
Solve the following differential equation.
y dx + (x  y^{2} ) dy = 0
Choose the correct alternative.
The integrating factor of `dy/dx  y = e^x `is e^{x}, then its solution is
Solve the differential equation:
`e^(dy/dx) = x`
Solve:
(x + y) dy = a^{2 }dx
x^{2}y dx – (x^{3} + y^{3}) dy = 0
`xy dy/dx = x^2 + 2y^2`
y dx – x dy + log x dx = 0
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
`yx ("d"y)/("d"x)` = x^{2} + 2y^{2}
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
State whether the following statement is True or False:
The integrating factor of the differential equation `("d"y)/("d"x)  y` = x is e^{–x}
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
Solve the following differential equation
sec^{2} x tan y dx + sec^{2} y tan x dy = 0
Solution: sec^{2} x tan y dx + sec^{2} y tan x dy = 0
∴ `(sec^2x)/tanx "d"x + square` = 0
Integrating, we get
`square + int (sec^2y)/tany "d"y` = log c
Each of these integral is of the type
`int ("f'"(x))/("f"(x)) "d"x` = log f(x) + log c
∴ the general solution is
`square + log tan y` = log c
∴ log tan x . tan y = log c
`square`
This is the general solution.
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: `("d"y)/("d"x) = cos(x + y) + sin(x + y)`. [Hint: Substitute x + y = z]
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.