The general solution of the differential equation 1 + x 2 + y 2 + x 2 y 2 + x y d y d x = 0 (where C is a constant of integration)

1 + y 2 + 1 + x 2 = 1 2 log e 1 + x 2 + 1 1 + x 2 - 1 + C

1 + y 2 + 1 + x 2 = 1 2 log e 1 + x 2 - 1 1 + x 2 + 1 + C

1 + y 2 - 1 + x 2 = 1 2 log e 1 + x 2 - 1 1 + x 2 + 1 + C

1 + y 2 - 1 + x 2 = 1 2 log e 1 + x 2 + 1 1 + x 2 - 1 + C

The solution curve of the differential equation, 1 + e - x 1 + y 2 d y d x = y 2 which passes through the point 0 , 1 , is

y 2 + 1 = y log e 1 + e - x 2 + 2

y 2 + 1 = y log e 1 + e x 2 + 2

Let b be a nonzero real number. Suppose f : ℝ → ℝ is a differentiable function such that f 0 = 1 . If the derivative f ' of f satisfies the equation f ' x = f x b 2 + x 2 for all x ∈ ℝ , then which of the following statements is/are TRUE?

f x f - x = 1 for all x ∈ ℝ

If b < 0 , then f is a decreasing function

f x - f - x = 0 for all x ∈ ℝ

Let f : ℝ → ℝ and g : ℝ → ℝ be functions satisfying f x + y = f x + f y + f x f y and f x = x g x for all x , y ∈ ℝ . If lim x → 0 g x = 1 , then which of the following statements is/are TRUE?

The derivative f ' 0 is equal to 1

The derivative f ' 1 is equal to 1

HARD

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A river of width $a$ is flowing towards positive $x$ axis. The velocity of current is directly proportional to product of distances from the two banks, constant of proportionality being $k$ . A boat is rowed with a velocity $u$ directly across the stream in positive $y$ direction. (Assume starting point as origin ) which of the following is true for the trajectory of the boat.

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Important Questions on Differential Equations

HARD

Mathematics>Integral Calculus>Differential Equations>Variable Separable Form

Let

y = y (x)

be the solution of the differential equation,

\frac{2 + \sin x}{y + 1} . \frac{d y}{d x} = - \cos x

y > 0, y (0) = 1

. If

y (π) = a

and

\frac{d y}{d x}

x = π

b

, then the ordered pair

(a, b)

is equal to

MEDIUM

Mathematics>Integral Calculus>Differential Equations>Variable Separable Form

A tangent to the curve,

y = f (x)

P (x, y)

meets

x

-axis at

A

and

y

-axis at

B

. If

A P : B P = 1 : 3

and

f (1) = 1,

then the curve also passes through the point

HARD

Mathematics>Integral Calculus>Differential Equations>Variable Separable Form

The particular solution of the differential equation

\log (\frac{d y}{d x}) = x

, when

x = 0

y = 1

is …..

MEDIUM

Mathematics>Integral Calculus>Differential Equations>Variable Separable Form

The general solution of the differential equation

\sqrt{1 + x^{2} + y^{2} + x^{2} y^{2}} + x y \frac{d y}{d x} = 0

(where C is a constant of integration)

EASY

Mathematics>Integral Calculus>Differential Equations>Variable Separable Form

The solution of

\frac{d y}{d x} = 2^{y - x}

is:

HARD

Mathematics>Integral Calculus>Differential Equations>Variable Separable Form

Let $f : (0, \infty) \to (0, \infty)$ be a differentiable function such that $f (1) = e$ and $\lim_{t \to x} \frac{t^{2} f^{2} (x) - x^{2} f^{2} (t)}{t - x} = 0 .$ If $f (x) = 1$ , then $x$ is equal to:

HARD

Mathematics>Integral Calculus>Differential Equations>Variable Separable Form

f (x)

is a non-zero polynomial of degree four, having local extreme points at

x = - 1, 0, 1;

then the set

S = {x \in R : f (x) = f (0)}

contains exactly

HARD

Mathematics>Integral Calculus>Differential Equations>Variable Separable Form

Solution of the differential equation

\frac{d y}{d x} = \sin (x + y) + \cos (x + y)

is equal to

HARD

Mathematics>Integral Calculus>Differential Equations>Variable Separable Form

Let the population of rabbits surviving at a time

t

be governed by the differential equation

\frac{d p (t)}{d t} = \frac{1}{2} {p (t) - 400}

. If

p (0) = 100

, then

p (t)

equals

MEDIUM

Mathematics>Integral Calculus>Differential Equations>Variable Separable Form

(2 + \sin x) \frac{d y}{d x} + (y + 1) \cos x = 0

and

y (0) = 1

, then

y (\frac{π}{2})

is equal to

HARD

Mathematics>Integral Calculus>Differential Equations>Variable Separable Form

The solution curve of the differential equation,

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

which passes through the point

(0, 1)

, is

MEDIUM

Mathematics>Integral Calculus>Differential Equations>Variable Separable Form

Let

f : R \to R

be a differentiable function with

f (0) = 0

. If

y = f (x)

satisfies the differential equation

\frac{d y}{d x} = (2 + 5 y) (5 y - 2) .

If the value of

\lim_{x \to - \infty} f (x) = λ

, then

10 λ

is equal to

HARD

Mathematics>Integral Calculus>Differential Equations>Variable Separable Form

Let

b

be a nonzero real number. Suppose

f : ℝ \to ℝ

is a differentiable function such that

f (0) = 1

. If the derivative

f^{'}

f

satisfies the equation

f^{'} (x) = \frac{f (x)}{b^{2} + x^{2}}

for all

x \in ℝ,

then which of the following statements is/are TRUE?

HARD

Mathematics>Integral Calculus>Differential Equations>Variable Separable Form

y = y (x)

is the solution of the differential equation

\frac{5 + e^{x}}{2 + y} \cdot \frac{d y}{d x} + e^{x} = 0

satisfying

y (0) = 1

then value of

y (\log_{e} 13)

HARD

Mathematics>Integral Calculus>Differential Equations>Variable Separable Form

Let $f : ℝ \to ℝ$ and $g : ℝ \to ℝ$ be functions satisfying $f (x + y) = f (x) + f (y) + f (x) f (y) and f (x) = x g (x)$ for all $x, y \in ℝ .$ If $\lim_{x \to 0} g (x) = 1,$ then which of the following statements is/are TRUE?

HARD

Mathematics>Integral Calculus>Differential Equations>Variable Separable Form

f^{'} (x) = f (x) + \int_{0}^{1} f (x) d x, f (0) = 1

, then

f (x) =

HARD

Mathematics>Integral Calculus>Differential Equations>Variable Separable Form

The solution of

\frac{d y}{d x} = \cos (x + y) + \sin (x + y)

, is

MEDIUM

Mathematics>Integral Calculus>Differential Equations>Variable Separable Form

The particular solution of the differential equation

x d y + 2 y d x = 0

, when

x = 2 & y = 1

HARD

Mathematics>Integral Calculus>Differential Equations>Variable Separable Form

y (x)

is the solution of the differential equation

(x + 2) \frac{d y}{d x} = x^{2} + 4 x - 9, x \neq - 2

and

y (0) = 0,

then

y (- 4)

is equal to

HARD

Mathematics>Integral Calculus>Differential Equations>Variable Separable Form

Let

f : R \to R

be a differentiable function with

f (0) = 1

and satisfying the equation

f (x + y) = f (x) f^{'} (y) + f^{'} (x) f (y)

for all

x, y \in R

. Then, the value of

\log_{e} (f (4))

Important Questions on Differential Equations

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