Given that the slope of the tangent to a curve y = y ( x ) at any point x , y is 2 y x 2 . If the curve passes through the centre of the circle x 2 + y 2 - 2 x - 2 y = 0 , then its equation is

x log e ⁡ | y | = 2 ( x - 1 )

x 2 log e ⁡ | y | = - 2 ( x - 1 )

x log e ⁡ | y | = - 2 ( x - 1 )

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The differential equation $\frac{d y}{d x} = \frac{x (1 + y^{2})}{y (1 + x^{2})}$ represents a family of:

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Important Questions on Differential Equations

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Mathematics>Integral Calculus>Differential Equations>Applications of Differential Equations

The rate of growth of bacteria is proportional to the number present. If initially there were $1000$ bacteria and the number doubles in $1$ hour. Then, the number of bacteria after $2 \frac{1}{2}$ hours are $($ Given $\sqrt{2} = 1.414)$

MEDIUM

Mathematics>Integral Calculus>Differential Equations>Applications of Differential Equations

In a certain culture of bacteria, the rate of increase is proportional to the number present. It is found that the number doubles in

4

hours. Then, the number of times the bacteria are increased in

12

hours is

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Mathematics>Integral Calculus>Differential Equations>Applications of Differential Equations

If the body cools from $135 ° C$ to $80 ° C$ at room temperature of $25 ° C$ in $60$ minutes, then the temperature of body after $2$ hours is

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Mathematics>Integral Calculus>Differential Equations>Applications of Differential Equations

The population

P = P (t)

at time

t

of a certain species follows the differential equation

\frac{d P}{d t} = 0.5 P - 450

. If

P (0) = 850

, then the time at which population becomes zero is:

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Mathematics>Integral Calculus>Differential Equations>Applications of Differential Equations

At present, a firm is manufacturing

2000

items. It is estimated that the rate of change of production

P

w.r.t. additional number of workers

x

is given by

\frac{d P}{d x} = 100 - 12 \sqrt{x}

. If the firm employs

25

more workers, then the new level of production of items is

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Mathematics>Integral Calculus>Differential Equations>Applications of Differential Equations

A curve having the condition that the slope of tangent at some point is two times the slope of the straight line joining the same point to the origin of co-ordinates, is a/an

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Mathematics>Integral Calculus>Differential Equations>Applications of Differential Equations

At any point on a curve, the slope of the tangent is equal to the sum of abscissa and the product of ordinate and abscissa of that point. If the curve passes through

(0, 1)

, then the equation of the curve is

MEDIUM

Mathematics>Integral Calculus>Differential Equations>Applications of Differential Equations

The population of a city increases at the rate

3 %

per year. If at time

t

the population of city is

p,

then find equation of

p

in time

t .

MEDIUM

Mathematics>Integral Calculus>Differential Equations>Applications of Differential Equations

The equation of the curve passing through the point

(1, 1)

such that the slope of the tangent at any point

(x, y)

is equal to the product of its co-ordinates is

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Mathematics>Integral Calculus>Differential Equations>Applications of Differential Equations

The equation of the curve whose slope at any point is equal to

y + 2 x

and which passes through the origin is

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Mathematics>Integral Calculus>Differential Equations>Applications of Differential Equations

Let slope of the tangent line to a curve at any point

P (x, y)

be given by

\frac{x y^{2} + y}{x}

. If the curve intersects the line

x + 2 y = 4

x = - 2

, then the value of

y

, for which the point

(3, y)

lies on the curve, is :

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Mathematics>Integral Calculus>Differential Equations>Applications of Differential Equations

Given that the slope of the tangent to a curve

y = y (x)

at any point

(x, y)

\frac{2 y}{x^{2}}

. If the curve passes through the centre of the circle

x^{2} + y^{2} - 2 x - 2 y = 0,

then its equation is

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Mathematics>Integral Calculus>Differential Equations>Applications of Differential Equations

If a curve passes through the origin and the slope of the tangent to it at any point

(x, y)

\frac{x^{2} - 4 x + y + 8}{x - 2},

then this curve also passes through the point:

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Mathematics>Integral Calculus>Differential Equations>Applications of Differential Equations

Water at

100 ° C

cools in

15

minutes to

75 ° C

in a room temperature of

25 ° C

. Then, the temperature of water after

30

minutes is

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Mathematics>Integral Calculus>Differential Equations>Applications of Differential Equations

The rate of growth of bacteria in a culture is proportional to the number of bacteria present and the bacteria count is

1000

at initial time

t = 0 .

The number of bacteria is increased by

20 %

2

hours. If the population of bacteria is

2000

after

\frac{k}{\log_{e} (\frac{6}{5})}

hours, then

{(\frac{k}{\log_{e} 2})}^{2}

is equal to:

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Mathematics>Integral Calculus>Differential Equations>Applications of Differential Equations

The family of curves that is orthogonal to

x y = c^{2}

is -

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Mathematics>Integral Calculus>Differential Equations>Applications of Differential Equations

A differential equation representing the family of parabolas with axis parallel to

y -

axis and whose length of latus rectum is the distance of the point

(2, - 3)

from the line

3 x + 4 y = 5

, is given by:

EASY

Mathematics>Integral Calculus>Differential Equations>Applications of Differential Equations

The curve passing through the point

(1, 2)

given that the slope of the tangent at any point

(x, y)

\frac{2 x}{y}

represents

EASY

Mathematics>Integral Calculus>Differential Equations>Applications of Differential Equations

The differential equation of the family of curves,

x^{2} = 4 b (y + b), b \in R

, is.

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Mathematics>Integral Calculus>Differential Equations>Applications of Differential Equations

Let $f : [\frac{1}{2}, 1] \to R$ (the set of all real numbers) be a positive, non-constant and differentiable function such that $f^{'} (x) < 2 f (x)$ and $f (\frac{1}{2}) = 1$ . Then the value of $\int_{1 / 2}^{1} f (x) d x$ lies in the interval:

The differential equation dydx=x(1+y2)y(1+x2) represents a family of:

Important Questions on Differential Equations

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The differential equation $\frac{d y}{d x} = \frac{x (1 + y^{2})}{y (1 + x^{2})}$ represents a family of: