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.

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

Verify that y = 4 sin 3x is a solution of the differential equation \[\frac{d^2 y}{d x^2} + 9y = 0\]

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

Verify that \[y = e^{m \cos^{- 1} x}\] satisfies the differential equation \[\left( 1 - x^2 \right)\frac{d^2 y}{d x^2} - x\frac{dy}{dx} - m^2 y = 0\]

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

\[\frac{dy}{dx} = x e^x - \frac{5}{2} + \cos^2 x\]

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

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

x cos² y dx = y cos² x dy

xy dy = (y − 1) (x + 1) dx

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

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

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

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

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

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

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

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

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

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

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).

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

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

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

(x + y) (dx − dy) = dx + dy

x² dy + y (x + y) dx = 0

(x² − y²) dx − 2xy dy = 0

(x + 2y) dx − (2x − y) dy = 0

Solve the following differential equations:
\[\frac{dy}{dx} = \frac{y}{x}\left\{ \log y - \log x + 1 \right\}\]

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

Solve the following initial value problem:-
\[x\frac{dy}{dx} - y = \log x, y\left( 1 \right) = 0\]

Solve the following initial value problem:-

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

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:
\[\frac{dy}{dx} + y \cot x = 4x\text{ cosec }x, y\left( \frac{\pi}{2} \right) = 0\]

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.

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 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.

Write the differential equation representing the family of straight lines y = Cx + 5, where C is an arbitrary constant.

The differential equation of the ellipse \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = C\] is

The differential equation
\[\frac{dy}{dx} + Py = Q y^n , n > 2\] can be reduced to linear form by substituting

Verify that the function y = e^−3x is a solution of the differential equation \[\frac{d^2 y}{d x^2} + \frac{dy}{dx} - 6y = 0.\]

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

y = e^x + 1 y'' − y' = 0

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

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).

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 solution of `1/"x" * "dy"/"dx" = tan^-1 "x"` is

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

Solution	D.E.
y = e^x	`dy/ dx= y`

Solve the following differential equation.

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

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.

xdx + 2y dx = 0

Solve the following differential equation.

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

Solve the following differential equation.

`(x + y) dy/dx = 1`

Solve the following differential equation.

y dx + (x - y² ) dy = 0

The differential equation of `y = k_1e^x+ k_2 e^-x` is ______.

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

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 xdx + 2ydy = 0

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

Solve the following differential equation

`yx ("d"y)/("d"x)` = x² + 2y²

Choose the correct alternative:

General solution of `y - x ("d"y)/("d"x)` = 0 is

The function y = e^x is solution ______ of differential equation

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

Solve the following differential equation

`y log y ("d"x)/("d"y) + x` = log y

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.

The integrating factor of the differential equation `"dy"/"dx" (x log x) + y` = 2logx is ______.

Solve: ydx – xdy = x²ydx.

`d/(dx)(tan^-1 (sqrt(1 + x^2) - 1)/x)` is equal to:

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

Advertisements

Advertisements

Question

Advertisements

Solution

APPEARS IN

RELATED QUESTIONS

Select a course

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

Advertisements

Advertisements

Question

Advertisements

SolutionShow Solution

APPEARS IN

RELATED QUESTIONS

Select a course

Solution