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

Let f : 0 , ∞ → R be a differentiable function such that f ' x = 2 - f x x ∀ x ∈ ( 0 , ∞ ) and f 1 ≠ 1 . Then,

lim x → 0 + ⁡ f ′ 1 x = 1

lim x → 0 + ⁡ x 2 f ′ x = 0

Let y ( x ) be a solution of the differential equation 1 + e x y ' + y e x = 1 . If y 0 = 2 , then which of the following statements is (are) true?

y ( x ) has a critical point in the interval ( - 1 , 0 )

y ( x ) has no critical point in the interval ( - 1 , 0 )

For any real numbers α and β , let y α , β x , x ∈ R , be the solution of the differential equation d y d x + α y = x e β x , y 1 = 1 Let S = y α , β x : α , β ∈ R . Then which of the following functions belong(s) to the set S ?

f x = e x 2 x - 1 2 + e - e 2 4 e - x

f x = - x 2 2 e - x + e + 1 2 e - x

f x = e x 2 1 2 - x + e + e 2 4 e - x

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

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 : 0 , ∞ → R be a differentiable function such that f ' x = 2 - f x x ∀ x ∈ ( 0 , ∞ ) and f 1 ≠ 1 . Then,

lim x → 0 + ⁡ f ′ 1 x = 1

lim x → 0 + ⁡ x 2 f ′ x = 0

HARD

JEE Advanced

IMPORTANT

Earn 100

If $y = y (x)$ 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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Important Questions on Differential Equations

HARD

JEE Advanced

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 28 - Differential Equations>JEE Advanced Paper 2 - 2014>Q 8

The function

y = f (x)

is the solution of the differential equation

\frac{d y}{d x} + \frac{x y}{x^{2} - 1} = \frac{x^{4} + 2 x}{\sqrt{1 - x^{2}}}

(- 1, 1)

satisfying

f (0) = 0 .

Then

\int_{- \frac{\sqrt{3}}{2}}^{\frac{\sqrt{3}}{2}} f (x) d x

HARD

JEE Advanced

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 28 - Differential Equations>JEE Advanced Paper 1 - 2018>Q 9

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?

HARD

JEE Advanced

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 28 - Differential Equations>JEE Advanced Paper 1 - 2016>Q 10

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

JEE Advanced

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 28 - Differential Equations>JEE Advanced Paper 1 - 2016>Q 11

Let

f : (0, \infty) \to R

be a differentiable function such that

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

\forall x \in (0, \infty)

and

f (1) \neq 1

. Then,

HARD

JEE Advanced

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 28 - Differential Equations>JEE Advanced Paper 1 - 2015>Q 12

Let

y (x)

be a solution of the differential equation

(1 + e^{x}) y^{'} + y e^{x} = 1 .

y (0) = 2

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

HARD

JEE Advanced

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 28 - Differential Equations>JEE Advanced Paper 1 - 2015>Q 13

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 ?

HARD

JEE Advanced

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 28 - Differential Equations>JEE Advanced Paper 1 - 2013>Q 14

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:

HARD

JEE Advanced

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 28 - Differential Equations>JEE Advanced Paper 1 - 2013>Q 15

A curve passes through the point

(1, \frac{π}{6})

. Let the slope of the curve at each point

(x, y)

\frac{y}{x} + \sec (\frac{y}{x}), x > 0

. Then the equation of the curve is

If y=yx satisfies the differential equation 8x9+xdy=4+9+x-1dx and x>0, y0=7, then y256=

Important Questions on Differential Equations

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If $y = y (x)$ 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 $y (256) =$