Choose the correct option from the given alternatives: The solution of xdydxx1x⋅dydx=tan-1x is - Mathematics and Statistics

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.

Find the differential equation of all the parabolas with latus rectum '4a' and whose axes are parallel to x-axis.

Show that the differential equation of which y = 2(x² − 1) + \[c e^{- x^2}\] is a solution, is \[\frac{dy}{dx} + 2xy = 4 x^3\]

Form the differential equation of the family of hyperbolas having foci on x-axis and centre at the origin.

Hence, the given function is the solution to the given differential equation. \[\frac{c - x}{1 + cx}\] is a solution of the differential equation \[(1+x^2)\frac{dy}{dx}+(1+y^2)=0\].

Differential equation \[\frac{dy}{dx} + y = 2, y \left( 0 \right) = 3\] Function y = e^−x + 2

Differential equation \[\frac{d^2 y}{d x^2} + y = 0, y \left( 0 \right) = 1, y' \left( 0 \right) = 1\] Function y = sin x + cos x

\[\frac{dy}{dx} + 2x = e^{3x}\]

\[\left( x^2 + 1 \right)\frac{dy}{dx} = 1\]

\[\frac{dy}{dx} = \tan^{- 1} x\]

\[\frac{dy}{dx} = \cos^3 x \sin^2 x + x\sqrt{2x + 1}\]

\[\left( x^3 + x^2 + x + 1 \right)\frac{dy}{dx} = 2 x^2 + x\]

\[\sin\left( \frac{dy}{dx} \right) = k ; y\left( 0 \right) = 1\]

\[\left( x - 1 \right)\frac{dy}{dx} = 2 xy\]

\[\frac{dy}{dx} = \left( e^x + 1 \right) y\]

Solve the differential equation \[\frac{dy}{dx} = e^{x + y} + x^2 e^y\].

\[x\frac{dy}{dx} + y = y^2\]

(1 − x²) dy + xy dx = xy² dx

tan y dx + sec² y tan x dy = 0

tan y \[\frac{dy}{dx}\] = sin (x + y) + sin (x − y)

\[\frac{dy}{dx} = 1 - x + y - xy\]

dy + (x + 1) (y + 1) dx = 0

\[\frac{dy}{dx} = \left( \cos^2 x - \sin^2 x \right) \cos^2 y\]

Solve the following differential equation:
(xy² + 2x) dx + (x² y + 2y) dy = 0

\[2x\frac{dy}{dx} = 5y, y\left( 1 \right) = 1\]

\[\frac{dy}{dx} = 2 e^{2x} y^2 , y\left( 0 \right) = - 1\]

\[\cos y\frac{dy}{dx} = e^x , y\left( 0 \right) = \frac{\pi}{2}\]

\[2\left( y + 3 \right) - xy\frac{dy}{dx} = 0\], y(1) = −2

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.

Find the particular solution of the differential equation
(1 – y²) (1 + log x) dx + 2xy dy = 0, given that y = 0 when x = 1.

\[\frac{dy}{dx} = \frac{\left( x - y \right) + 3}{2\left( x - y \right) + 5}\]

\[\left( x + y \right)^2 \frac{dy}{dx} = 1\]

\[\left( x + y + 1 \right)\frac{dy}{dx} = 1\]

\[x\frac{dy}{dx} = x + y\]

\[x^2 \frac{dy}{dx} = x^2 - 2 y^2 + xy\]

\[x^2 \frac{dy}{dx} = x^2 + xy + y^2 \]

3x² dy = (3xy + y²) dx

\[x\frac{dy}{dx} = y - x \cos^2 \left( \frac{y}{x} \right)\]

Find the particular solution of the differential equation \[\frac{dy}{dx} = \frac{xy}{x^2 + y^2}\] given that y = 1 when x = 0.

Solve the following initial value problem:-

\[y' + y = e^x , y\left( 0 \right) = \frac{1}{2}\]

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 \[x = \frac{\pi}{2}\]

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?

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.

At every point on a curve the slope is the sum of the abscissa and the product of the ordinate and the abscissa, and the curve passes through (0, 1). Find the equation of the curve.

Find the equation of the curve which passes through the point (1, 2) and the distance between the foot of the ordinate of the point of contact and the point of intersection of the tangent with x-axis is twice the abscissa of the point of contact.

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.

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 solution of the differential equation \[\frac{dy}{dx} - \frac{y\left( x + 1 \right)}{x} = 0\] is given by

The solution of the differential equation y₁ y₃ = y₂² is

Which of the following differential equations has y = C₁ e^x + C₂ e⁻^x as the general solution?

The integrating factor of the differential equation \[x\frac{dy}{dx} - y = 2 x^2\]

Solve the following differential equation : \[y^2 dx + \left( x^2 - xy + y^2 \right)dy = 0\] .

Solve the following differential equation : \[\left( \sqrt{1 + x^2 + y^2 + x^2 y^2} \right) dx + xy \ dy = 0\].

Show that y = ae^2x + be^−x is a solution of the differential equation \[\frac{d^2 y}{d x^2} - \frac{dy}{dx} - 2y = 0\]

y² dx + (x² − xy + y²) dy = 0

Form the differential equation representing the family of parabolas having vertex at origin and axis along positive direction of x-axis.

If a + ib = `("x" + "iy")/("x" - "iy"),` prove that `"a"^2 +"b"^2 = 1` and `"b"/"a" = (2"xy")/("x"^2 - "y"^2)`

In the following example, verify that the given function is a solution of the corresponding differential equation.

Solution	D.E.
y = xn	`x^2(d^2y)/dx^2 - n xx (xdy)/dx + ny =0`

Determine the order and degree of the following differential equations.

Solution	D.E
y = ae^x+ be^−x	`(d^2y)/dx^2= 1`

Solve the following differential equation.

`y^3 - dy/dx = x dy/dx`

Solve the following differential equation.

`dy /dx +(x-2 y)/ (2x- y)= 0`

Solve the following differential equation.

dr + (2r)dθ= 8dθ

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.

Solve the differential equation:

dr = a r dθ − θ dr

Solve the differential equation `("d"y)/("d"x) + y` = e^−x

Choose the correct alternative:

Differential equation of the function c + 4yx = 0 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

sec² x tan y dx + sec² y tan x dy = 0

Solution: sec² x tan y dx + sec² 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 + c₁

∴ e^–2y = – 2sin x – 2c₁

∴ `square` = c, where c = – 2c₁

This is general solution.

When x = `pi/6`, y = 0, we have

`"e"^0 + 2sin pi/6` = c

∴ c = `square`

∴ particular solution is `square`

Integrating factor of the differential equation `x "dy"/"dx" - y` = sinx is ______.

Integrating factor of the differential equation `"dy"/"dx" - y` = cos x is e^x.

Given that `"dy"/"dx" = "e"^-2x` and y = 0 when x = 5. Find the value of x when y = 3.

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?

Choose the correct option from the given alternatives: The solution of xdydxx1x⋅dydx=tan-1x is - Mathematics and Statistics

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Choose the correct option from the given alternatives: The solution of xdydxx1x⋅dydx=tan-1x is - Mathematics and Statistics

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