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

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

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

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

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

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

\[\frac{dy}{dx} = x^5 + x^2 - \frac{2}{x}, x \neq 0\]

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

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

\[\sin^4 x\frac{dy}{dx} = \cos x\]

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

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

(e^y + 1) cos x dx + e^y sin x dy = 0

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

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

tan y dx + sec² y tan x dy = 0

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

(y + xy) dx + (x − xy²) dy = 0

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

Solve the following differential equation:
\[\text{ cosec }x \log y \frac{dy}{dx} + x^2 y^2 = 0\]

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

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

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

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

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

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

Solve the differential equation \[x\frac{dy}{dx} + \cot y = 0\] 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.

Find the solution of the differential equation cos y dy + cos x sin y dx = 0 given that y = \[\frac{\pi}{2}\], when x = \[\frac{\pi}{2}\]

In a bank principal increases at the rate of r% per year. Find the value of r if ₹100 double itself in 10 years (log_e 2 = 0.6931).

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).

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

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

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

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

(x + y) (dx − dy) = dx + dy

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

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

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

(y² − 2xy) dx = (x² − 2xy) dy

Solve the following initial value problem:-

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

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:-

\[\frac{dy}{dx} + 2y \tan x = \sin x; y = 0\text{ when }x = \frac{\pi}{3}\]

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

A population grows at the rate of 5% per year. How long does it take for the population to double?

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 equation of the curve which passes through the point (2, 2) and satisfies the differential equation
\[y - x\frac{dy}{dx} = y^2 + \frac{dy}{dx}\]

Find the equation of the curve passing through the point \[\left( 1, \frac{\pi}{4} \right)\] and tangent at any point of which makes an angle tan⁻¹\[\left( \frac{y}{x} - \cos^2 \frac{y}{x} \right)\] with x-axis.

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.

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.

Define a differential equation.

Write the differential equation obtained eliminating the arbitrary constant C in the equation xy = C².

The integrating factor of the differential equation (x log x)
\[\frac{dy}{dx} + y = 2 \log x\], is given by

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

The differential equation \[x\frac{dy}{dx} - y = x^2\], has the general solution

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

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

If x^myⁿ = (x + y)^m+n, prove that \[\frac{dy}{dx} = \frac{y}{x} .\]

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

In the following verify that the given functions (explicit or implicit) is a solution of the corresponding differential equation:-

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

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).

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

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

Determine the order and degree of the following differential equations.

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

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.

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

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.

`dy/dx + y` = 3

Solve the following differential equation.

`dy/dx + 2xy = x`

Solve the following differential equation.

`(x + a) dy/dx = – y + a`

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

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

Choose the correct alternative.

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

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.

State whether the following is True or False:

The integrating factor of the differential equation `dy/dx - y = x` is e^-x

Solve

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

Solve the differential equation sec²y tan x dy + sec²x tan y dx = 0

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

Solve: `("d"y)/("d"x) + 2/xy` = x²

Solve the following differential equation

`x^2 ("d"y)/("d"x)` = x² + xy − y²

Choose the correct alternative:

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

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

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

Verify y = log x + c is the solution of differential equation `x ("d"^2y)/("d"x^2) + ("d"y)/("d"x)` = 0

Verify y = `a + b/x` is solution of `x(d^2y)/(dx^2) + 2 (dy)/(dx)` = 0

y = `a + b/x`

`(dy)/(dx) = square`

`(d^2y)/(dx^2) = square`

Consider `x(d^2y)/(dx^2) + 2(dy)/(dx)`

= `x square + 2 square`

= `square`

Hence y = `a + b/x` is solution of `square`

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

Solution of `x("d"y)/("d"x) = y + x tan y/x` is `sin(y/x)` = cx

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

The value of `dy/dx` if y = |x – 1| + |x – 4| at x = 3 is ______.

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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