Solve the differential equation xdx + 2ydy = 0

Verify that y = log \[\left( x + \sqrt{x^2 + a^2} \right)^2\] satisfies the differential equation \[\left( a^2 + x^2 \right)\frac{d^2 y}{d x^2} + x\frac{dy}{dx} = 0\]

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

For the following differential equation verify that the accompanying function is a solution:

Differential equation	Function
\[x + y\frac{dy}{dx} = 0\]	\[y = \pm \sqrt{a^2 - x^2}\]

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

Differential equation \[\frac{d^2 y}{d x^2} - 3\frac{dy}{dx} + 2y = 0, y \left( 0 \right) = 1, y' \left( 0 \right) = 3\] Function y = e^x + e^2x

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

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

\[\cos x\frac{dy}{dx} - \cos 2x = \cos 3x\]

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

\[\sqrt{a + x} dy + x\ dx = 0\]

\[\frac{dy}{dx} + \frac{1 + y^2}{y} = 0\]

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

\[5\frac{dy}{dx} = e^x y^4\]

x cos y dy = (xe^x log x + e^x) dx

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

tan y dx + sec² y tan x dy = 0

\[x\sqrt{1 - y^2} dx + y\sqrt{1 - x^2} dy = 0\]

y (1 + e^x) dy = (y + 1) e^x dx

(y² + 1) dx − (x² + 1) dy = 0

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

Solve the following differential equation:
\[xy\frac{dy}{dx} = 1 + x + y + xy\]

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

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

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

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

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

Solve the differential equation \[x\frac{dy}{dx} + \cot y = 0\] given that \[y = \frac{\pi}{4}\], when \[x=\sqrt{2}\]

Find the particular solution of e^dy^/dx = x + 1, given that y = 3, when x = 0.

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.

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

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

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

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

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

(y² − 2xy) dx = (x² − 2xy) dy

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

Solve the following initial value problem:-

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

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.

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 normal to a given curve at each point (x, y) on the curve passes through the point (3, 0). If the curve contains the point (3, 4), find its equation.

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.

The slope of a curve at each of its points is equal to the square of the abscissa of the point. Find the particular curve through the point (−1, 1).

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

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 \[\frac{du}{dx} + P\left( x \right) u = Q\left( x \right)\]

What is integrating factor of \[\frac{dy}{dx}\] + y sec x = tan x?

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

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

y² dx + (x² − xy + y²) dy = 0

Form the differential equation representing the family of curves y = a sin (x + b), where a, b are arbitrary constant.

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 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).

Determine the order and degree of the following differential equations.

Solution	D.E.
ax2 + by2 = 5	`xy(d^2y)/dx^2+ x(dy/dx)^2 = y dy/dx`

Solve the following differential equation.

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

Solve the following differential equation.

y dx + (x - y² ) dy = 0

Choose the correct alternative.

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

Choose the correct alternative.

The integrating factor of `dy/dx - y = e^x `is e^x, then its solution is

Solve the following differential equation `("d"y)/("d"x)` = x²y + y

Solve: `("d"y)/("d"x) + 2/xy` = x²

Choose the correct alternative:

Solution of the equation `x("d"y)/("d"x)` = y log y 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 = cx is the solution of differential equation `("d"y)/("d"x) = y/x`

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

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

Solve the differential equation `"dy"/"dx"` = 1 + x + y² + xy², when y = 0, x = 0.

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)` =

Which of the following is an example of an ordinary differential equation?

Solve the differential equation xdx + 2ydy = 0

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Solve the differential equation xdx + 2ydy = 0

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