Solve the differential equation (x2 – yx2)dy + (y2 + xy2)dx = 0 - 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.

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.

Show that Ax² + By² = 1 is a solution of the differential equation x \[\left\{ y\frac{d^2 y}{d x^2} + \left( \frac{dy}{dx} \right)^2 \right\} = y\frac{dy}{dx}\]

Verify that y² = 4a (x + a) is a solution of the differential equations
\[y\left\{ 1 - \left( \frac{dy}{dx} \right)^2 \right\} = 2x\frac{dy}{dx}\]

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

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

Function y = e^x + 1

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

Differential equation \[\frac{d^2 y}{d x^2} - y = 0, y \left( 0 \right) = 2, y' \left( 0 \right) = 0\] Function y = e^x + e⁻^x

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

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

\[\left( x + 2 \right)\frac{dy}{dx} = x^2 + 3x + 7\]

(sin x + cos x) dy + (cos x − sin x) dx = 0

\[\frac{dy}{dx} - x \sin^2 x = \frac{1}{x \log x}\]

\[\frac{dy}{dx} = x^5 \tan^{- 1} \left( x^3 \right)\]

\[\sqrt{1 - x^4} dy = x\ dx\]

\[\frac{dy}{dx} = x \log x\]

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

\[\frac{dy}{dx} = \frac{1 - \cos 2y}{1 + \cos 2y}\]

(1 + x²) dy = xy dx

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

xy (y + 1) dy = (x² + 1) dx

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

\[x\frac{dy}{dx} + \cot y = 0\]

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

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

\[\cos x \cos y\frac{dy}{dx} = - \sin x \sin y\]

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

(y² + 1) dx − (x² + 1) dy = 0

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

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

Solve the following differential equation:
\[\left( 1 + y^2 \right) \tan^{- 1} xdx + 2y\left( 1 + x^2 \right)dy = 0\]

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

\[\frac{dy}{dx} = 2 e^x y^3 , y\left( 0 \right) = \frac{1}{2}\]

\[\frac{dr}{dt} = - rt, r\left( 0 \right) = r_0\]

\[\frac{dy}{dx} = y \tan x, y\left( 0 \right) = 1\]

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

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

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

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

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

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

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

\[\frac{dy}{dx} + 1 = e^{x + y}\]

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

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

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

y e^x^/y dx = (xe^x^/y + y) dy

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:
\[x\frac{dy}{dx} + y = x \cos x + \sin x, y\left( \frac{\pi}{2} \right) = 1\]

Solve the following initial value problem:-

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

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 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?

A bank pays interest by continuous compounding, that is, by treating the interest rate as the instantaneous rate of change of principal. Suppose in an account interest accrues at 8% per year, compounded continuously. Calculate the percentage increase in such an account over one year.

Find the curve for which the intercept cut-off by a tangent on x-axis is equal to four times the ordinate of the point of contact.

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.

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.

Radium decomposes at a rate proportional to the quantity of radium present. It is found that in 25 years, approximately 1.1% of a certain quantity of radium has decomposed. Determine approximately how long it will take for one-half of the original amount of radium to decompose?

The solution of the differential equation \[\frac{dy}{dx} = \frac{ax + g}{by + f}\] represents a circle when

The integrating factor of the differential equation \[\left( 1 - y^2 \right)\frac{dx}{dy} + yx = ay\left( - 1 < y < 1 \right)\] is ______.

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

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

Find the coordinates of the centre, foci and equation of directrix of the hyperbola x² – 3y² – 4x = 8.

Solve the differential equation:

`"x"("dy")/("dx")+"y"=3"x"^2-2`

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

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

Solution	D.E.
xy = log y + k	y' (1 - xy) = y²

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`

Solve the following differential equation.

`dy/dx = x^2 y + y`

Solve the following differential equation.

`(dθ)/dt = − k (θ − θ_0)`

For each of the following differential equations find the particular solution.

`y (1 + logx)dx/dy - x log x = 0`,

when x=e, y = e².

Solve the following differential equation.

y² dx + (xy + x² ) dy = 0

Solve the following differential equation.

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

Solve the following differential equation.

(x² − y²) dx + 2xy 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.

`(x + a) dy/dx = – y + a`

Choose the correct alternative.

The differential equation of y = `k_1 + k_2/x` is

The solution of `dy/ dx` = 1 is ______

The solution of `dy/dx + x^2/y^2 = 0` is ______

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`

Select and write the correct alternative from the given option for the question

The differential equation of y = Ae^5x + Be^–5x is

Solve the differential equation xdx + 2ydy = 0

Solve the following differential equation

`yx ("d"y)/("d"x)` = x² + 2y²

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 y²dx + (xy + x²) dy = 0

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`

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`

Given that `"dy"/"dx"` = ye^x and x = 0, y = e. Find the value of y when x = 1.

An appropriate substitution to solve the differential equation `"dx"/"dy" = (x^2 log(x/y) - x^2)/(xy log(x/y))` is ______.

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

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

The differential equation of all non horizontal lines in a plane is `("d"^2x)/("d"y^2)` = 0

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

Solve the differential equation

`x + y dy/dx` = x² + y²

Solve the differential equation (x2 – yx2)dy + (y2 + xy2)dx = 0 - Mathematics and Statistics

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Solve the differential equation (x2 – yx2)dy + (y2 + xy2)dx = 0 - Mathematics and Statistics

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