The solution of the differential equation 1 + x x 2 + y 2 d x + x 2 + y 2 - 1 y d y = 0 is

x - y 2 2 + 1 3 x 2 + y 2 3 2 = C

x 2 + y 2 2 + 1 3 x 2 + y 2 3 2 = C

x - y 2 3 + 1 2 x 2 + y 2 1 2 = C

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For any differentiable function $y = f (x)$ , the value of $\frac{d^{2} y}{d x^{2}} + {(\frac{d y}{d x})}^{3} \frac{d^{2} x}{d y^{2}}$ is:

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Important Questions on Differential Equation

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Mathematics>Integral Calculus>Differential Equation>Basics of Differential Equations

Let

y = y (x)

be a solution of the differential equation,

\sqrt{1 - x^{2}} \frac{d y}{d x} + \sqrt{1 - y^{2}} = 0, |x| < 1 .

y (\frac{1}{2}) = \frac{\sqrt{3}}{2},

then

y (\frac{- 1}{\sqrt{2}})

is equal to

MEDIUM

Mathematics>Integral Calculus>Differential Equation>Basics of Differential Equations

The differential equation whose linearly independent solutions are

c o s 2 x, s i n 2 x, e^{- x},

MEDIUM

Mathematics>Integral Calculus>Differential Equation>Basics of Differential Equations

y = y (x)

is the solution of the differential equation,

e^{y} (\frac{d y}{d x} - 1) = e^{x}

such that

y (0) = 0,

then

y (1)

is equal to

HARD

Mathematics>Integral Calculus>Differential Equation>Basics of Differential Equations

The differential equation of the family of circles with fixed radius

5

units and center on the line

y = 2

MEDIUM

Mathematics>Integral Calculus>Differential Equation>Basics of Differential Equations

The differential equation of all parabolas whose axis is

y - a x i s

EASY

Mathematics>Integral Calculus>Differential Equation>Basics of Differential Equations

The order of the differential equation

y = C_{1} e^{C_{2} + x} + C_{3} e^{C_{4} + x}

HARD

Mathematics>Integral Calculus>Differential Equation>Basics of Differential Equations

The general solution of the differential equation

(y^{2} - x^{3}) d x - x y d y = 0, (x \neq 0)

is (where

c

is a constant of integration)

EASY

Mathematics>Integral Calculus>Differential Equation>Basics of Differential Equations

y = (\frac{2}{π} x - 1) cosec x

is the solution of the differential equation,

\frac{d y}{d x} + p (x) y = - \frac{2}{π} cosec x, 0 < x < \frac{π}{2}

, then the function

p (x)

is equal to :

EASY

Mathematics>Integral Calculus>Differential Equation>Basics of Differential Equations

The differential equation of the family of curves

y = e^{x} (A \cos x + B \sin x)

,

where

A

and

B

are arbitrary constants is

MEDIUM

Mathematics>Integral Calculus>Differential Equation>Basics of Differential Equations

If the differential equation representing the family of all circles touching $x$ -axis at the origin is $(x^{2} - y^{2}) \frac{d y}{d x} = g (x) y$ , then $g (x)$ equals

EASY

Mathematics>Integral Calculus>Differential Equation>Basics of Differential Equations

The order and degree of the differential equation

\frac{d^{2} y}{{dx}^{2}} + \sqrt{x^{2} + {(\frac{dy}{dx})}^{3 / 2}} = 0

are respectively

MEDIUM

Mathematics>Integral Calculus>Differential Equation>Basics of Differential Equations

The differential equation representing the family of ellipses having foci either on the

x

-axis or on the

y

-axis, center at the origin and passing through the point

(0, 3)

EASY

Mathematics>Integral Calculus>Differential Equation>Basics of Differential Equations

Elimination of arbitrary constants

A

and

B

from

y = \frac{A}{x} + B, x > 0

leads to the differential equation

EASY

Mathematics>Integral Calculus>Differential Equation>Basics of Differential Equations

The degree of the differential equation $\frac{d^{2} y}{d x^{2}} + 3 {(\frac{d y}{d x})}^{2} = x^{2} \log (\frac{d^{2} y}{d x^{2}})$ is

HARD

Mathematics>Integral Calculus>Differential Equation>Basics of Differential Equations

x^{3} d y + x y \cdot d x = x^{2} d y + 2 y d x; y (2) = e

and

x > 1

, then

y (4)

is equal to :

MEDIUM

Mathematics>Integral Calculus>Differential Equation>Basics of Differential Equations

The solution of the differential equation

\{1 + x \sqrt{(x^{2} + y^{2})}\} d x + \{\sqrt{(x^{2} + y^{2})} - 1\} y d y = 0

EASY

Mathematics>Integral Calculus>Differential Equation>Basics of Differential Equations

The order and degree of the differential equation

{(1 + 3 \frac{d y}{d x})}^{\frac{2}{3}} = 4 \frac{d^{3} y}{d x^{3}}

are

HARD

Mathematics>Integral Calculus>Differential Equation>Basics of Differential Equations

y = y (x)

satisfies the differential equation

8 \sqrt{x} (\sqrt{9 + \sqrt{x}}) d y = {(\sqrt{4 + \sqrt{9 + \sqrt{x}}})}^{- 1} d x

and

x > 0, y (0) = \sqrt{7},

then

y (256) =

MEDIUM

Mathematics>Integral Calculus>Differential Equation>Basics of Differential Equations

The solution of the differential equation

\frac{d y}{d x} = \tan (\frac{y}{x}) + \frac{y}{x}

EASY

Mathematics>Integral Calculus>Differential Equation>Basics of Differential Equations

The order and degree of

{\{1 + {(\frac{d y}{d x})}^{2}\}}^{\frac{1}{2}} = {(\frac{d^{2} y}{d x^{2}})}^{2}

For any differentiable function y=fx, the value of d2ydx2+dydx3d2xdy2 is:

Important Questions on Differential Equation

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For any differentiable function $y = f (x)$ , the value of $\frac{d^{2} y}{d x^{2}} + {(\frac{d y}{d x})}^{3} \frac{d^{2} x}{d y^{2}}$ is: