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The order of the differential equation of all tangent lines to the parabola $y = x^{2}$ is

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Important Questions on Differential Equations

HARD

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

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

EASY

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

The order and degree of the differential equation whose solution is

y = c x + c^{2} - 3 c^{\frac{3}{2}} + 2,

where

c

is a parameter is

HARD

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

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 Equations>Basics of Differential Equation

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

MEDIUM

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

The solution of the differential equation

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

MEDIUM

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

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 Equations>Basics of Differential Equation

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) =

EASY

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

Elimination of arbitrary constants

A

and

B

from

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

leads to the differential equation

MEDIUM

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

The differential equation of all parabolas whose axis is

y - a x i s

EASY

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

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

EASY

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

The order of the differential equation

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

EASY

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

The order and degree of

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

HARD

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

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 Equations>Basics of Differential Equation

The differential equation whose linearly independent solutions are

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

EASY

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

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

EASY

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

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 :

HARD

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

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 Equations>Basics of Differential Equation

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

MEDIUM

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

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 Equations>Basics of Differential Equation

The order of the differential equation of all parabolas, whose latus rectum is

4 a

and axis parallel to the

x

-axis, is

The order of the differential equation of all tangent lines to the parabola y=x2 is

Important Questions on Differential Equations

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The order of the differential equation of all tangent lines to the parabola $y = x^{2}$ is