Advertisements
Advertisements
Choose the correct alternative:
Solution of the equation `x("d"y)/("d"x)` = y log y is
Options
y = ae^{x}
y = be^{2x}
y = be^{–2x}
y = e^{ax}
Advertisements
Solution
y = e^{ax}
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`
Verify that y^{2} = 4ax is a solution of the differential equation y = x \[\frac{dy}{dx} + a\frac{dx}{dy}\]
Show that y = e^{−x} + ax + b is solution of the differential equation\[e^x \frac{d^2 y}{d x^2} = 1\]
For the following differential equation verify that the accompanying function is a solution:
Differential equation  Function 
\[y = \left( \frac{dy}{dx} \right)^2\]

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

(e^{y} + 1) cos x dx + e^{y} sin x dy = 0
Solve the following differential equation:
(xy^{2} + 2x) dx + (x^{2} y + 2y) dy = 0
Solve the following differential equation:
\[xy\frac{dy}{dx} = 1 + x + y + xy\]
Solve the following differential equation:
\[y\left( 1  x^2 \right)\frac{dy}{dx} = x\left( 1 + y^2 \right)\]
Solve the following differential equation:
\[y e^\frac{x}{y} dx = \left( x e^\frac{x}{y} + y^2 \right)dy, y \neq 0\]
The volume of a spherical balloon being inflated changes at a constant rate. If initially its radius is 3 units and after 3 seconds it is 6 units. Find the radius of the balloon after `t` seconds.
x^{2} dy + y (x + y) dx = 0
(x^{2} − y^{2}) dx − 2xy dy = 0
y e^{x}^{/y} dx = (xe^{x}^{/y} + y) dy
Solve the following initial value problem:
\[y' + y = e^x , y\left( 0 \right) = \frac{1}{2}\]
Solve the following initial value problem:
\[x\frac{dy}{dx}  y = \left( x + 1 \right) e^{ x} , y\left( 1 \right) = 0\]
If the interest is compounded continuously at 6% per annum, how much worth Rs 1000 will be after 10 years? How long will it take to double Rs 1000?
Find the equation of the curve which passes through the point (2, 2) and satisfies the differential equation
\[y  x\frac{dy}{dx} = y^2 + \frac{dy}{dx}\]
Show that the equation of the curve whose slope at any point is equal to y + 2x and which passes through the origin is y + 2 (x + 1) = 2e^{2x}.
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).
Show that all curves for which the slope at any point (x, y) on it is \[\frac{x^2 + y^2}{2xy}\] are rectangular hyperbola.
The xintercept of the tangent line to a curve is equal to the ordinate of the point of contact. Find the particular curve through the point (1, 1).
Which of the following is the integrating factor of (x log x) \[\frac{dy}{dx} + y\] = 2 log x?
If x^{m}y^{n} = (x + y)^{m+n}, prove that \[\frac{dy}{dx} = \frac{y}{x} .\]
Form the differential equation of the family of circles having centre on yaxis and radius 3 unit.
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.
Find the particular solution of the differential equation `"dy"/"dx" = "xy"/("x"^2+"y"^2),`given that y = 1 when x = 0
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
Find the differential equation whose general solution is
x^{3} + y^{3} = 35ax.
Solve the following differential equation.
`y^3  dy/dx = x dy/dx`
For each of the following differential equations 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.
`xy dy/dx = x^2 + 2y^2`
Solve the following differential equation.
`dy/dx + y` = 3
Solve the following differential equation.
`(x + y) dy/dx = 1`
Solve the following differential equation.
`dy/dx + 2xy = x`
Choose the correct alternative.
The solution of `x dy/dx = y` log y is
Choose the correct alternative.
Bacteria increases at the rate proportional to the number present. If the original number M doubles in 3 hours, then the number of bacteria will be 4M in
State whether the following is True or False:
The integrating factor of the differential equation `dy/dx  y = x` is e^{x}
State whether the following is True or False:
The degree of a differential equation is the power of the highest ordered derivative when all the derivatives are made free from negative and/or fractional indices if any.
Solve the differential equation:
`e^(dy/dx) = x`
Solve:
(x + y) dy = a^{2 }dx
Solve
`dy/dx + 2/ x y = x^2`
x^{2}y dx – (x^{3} + y^{3}) dy = 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:
Differential equation of the function c + 4yx = 0 is
A solution of differential equation which can be obtained from the general solution by giving particular values to the arbitrary constant is called ______ solution
The function y = e^{x} is solution ______ of differential equation
The solution of differential equation `x^2 ("d"^2y)/("d"x^2)` = 1 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}
Solve the following differential equation `("d"y)/("d"x)` = x^{2}y + y
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.
Find the particular solution of the following differential equation
`("d"y)/("d"x)` = e^{2y} cos x, when x = `pi/6`, y = 0.
Solution: The given D.E. is `("d"y)/("d"x)` = e^{2y} cos x
∴ `1/"e"^(2y) "d"y` = cos x dx
Integrating, we get
`int square "d"y` = cos x dx
∴ `("e"^(2y))/(2)` = sin x + c_{1}
∴ e^{–2y} = – 2sin x – 2c_{1}
∴ `square` = c, where c = – 2c_{1 }
This is general solution.
When x = `pi/6`, y = 0, we have
`"e"^0 + 2sin pi/6` = c
∴ c = `square`
∴ particular solution is `square`
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.