A solution curve of differential equation x 2 + x y + 4 x + 2 y + 4 d y d x - y 2 = 0 , x > 0 passes through the point ( 1 , 3 ) . Then the solution curve -

intersects y = x + 2 exactly at two points

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The differential equation of family of circles $x^{2} + y^{2} + 2 λ x + c = 0$ , where $λ, c$ both are parameters

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Important Questions on Differential Equations

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Mathematics>Integral Calculus>Differential Equations>Formation 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 Equations>Formation of Differential Equations

The differential equation of all parabolas whose axis is

y - a x i s

EASY

Mathematics>Integral Calculus>Differential Equations>Formation of Differential Equations

The order and degree of the differential equation

{(y^{‴})}^{2} + {(y^{″})}^{3} - {(y^{'})}^{4} + y^{5} = 0

HARD

Mathematics>Integral Calculus>Differential Equations>Formation of Differential Equations

A curve amongst the family of curves represented by the differential equation,

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

which passes through

(1, 1)

, is

HARD

Mathematics>Integral Calculus>Differential Equations>Formation of Differential Equations

The solution of the differential equation

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

x = f (y)

. If

f (- 1) = 1,

then

f (1)

is equal to

HARD

Mathematics>Integral Calculus>Differential Equations>Formation of Differential Equations

If a curve

y = f (x)

passes through the point

(1, - 1)

and satisfies the differential equation,

y (1 + x y) d x = x d y,

then

f (- \frac{1}{2})

is equal to

EASY

Mathematics>Integral Calculus>Differential Equations>Formation 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

HARD

Mathematics>Integral Calculus>Differential Equations>Formation 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 Equations>Formation 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}

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Mathematics>Integral Calculus>Differential Equations>Formation 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)

HARD

Mathematics>Integral Calculus>Differential Equations>Formation of Differential Equations

Let

f : (0, π) \to R

be a twice differentiable function such that

\lim_{t \to x} \{\frac{f (x) \sin t - f (t) \sin x}{t - x}\} = \sin^{2} x

for all

x \in (0, π)

. If

f (\frac{π}{6}) = - \frac{π}{12}

, then which of the following statement(s) is (are) TRUE ?

HARD

Mathematics>Integral Calculus>Differential Equations>Formation of Differential Equations

A solution curve of differential equation

(x^{2} + x y + 4 x + 2 y + 4) \frac{d y}{d x} - y^{2} = 0, x > 0

passes through the point

(1, 3)

. Then the solution curve -

HARD

Mathematics>Integral Calculus>Differential Equations>Formation of Differential Equations

Consider the family of all circles whose centers lie on the straight line

y = x .

If this family of circles is represented by the differential equation

P y^{"} + Q y^{'} + 1 = 0,

where

P, Q

are functions of

x, y & y^{'}

(y^{'} = \frac{d y}{d x}, y^{"} = \frac{d^{2} y}{d x^{2}})

, then which of the following statements is (are) true ?

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Mathematics>Integral Calculus>Differential Equations>Formation 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

HARD

Mathematics>Integral Calculus>Differential Equations>Formation 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) =

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Mathematics>Integral Calculus>Differential Equations>Formation 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 Equations>Formation of Differential Equations

The solution of the differential equation

y d x - x d y = x y d x

is ….

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Mathematics>Integral Calculus>Differential Equations>Formation 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 Equations>Formation of Differential Equations

What is the general solution of the differential equation

y d x - (x + 2 y^{2}) d y = 0 ?

HARD

Mathematics>Integral Calculus>Differential Equations>Formation of Differential Equations

Let

f : R \to R a n d g : R \to R

be two non-constant differentiable functions. If

f^{'} (x) = (e^{(f (x)) - g (x)}) g^{'} (x) f o r a l l x \in R

, and

f (1) = g (2) = 1

then which of the following statement(s) is (are) TRUE?

The differential equation of family of circles x2+y2+2λx+c=0, where λ,c both are parameters

Important Questions on Differential Equations

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The differential equation of family of circles $x^{2} + y^{2} + 2 λ x + c = 0$ , where $λ, c$ both are parameters