How many questions from Differential Equations appear in JEE?

Differential Equations has appeared in 167 of the last 186 JEE Main and JEE Advanced papers — about 90% of them — contributing 240 questions in total across those papers.

Is Differential Equations an important chapter for JEE?

Judged by how often it is actually tested, it appears in roughly 90% of papers. Chapters above about 50% are effectively guaranteed to show up every session, so they repay thorough preparation; lower-frequency chapters are better treated as targeted revision.

Where do these Differential Equations questions come from?

Every question is from an official JEE Main or JEE Advanced paper, transcribed from the original paper and tagged to this chapter. Answers follow the official answer key.

Differential Equations JEE Previous Year Questions with Answers

Q1·MathematicsSingle correctJEE Advanced 2026

Let

y : (-\infty, \infty) \to (0, \infty)

be the solution of the differential equation

\dfrac{dy}{dx} = \dfrac{e^{5x} y^{3} + y^{3}}{e^{x} + e^{x} y^{4}}

, satisfying

y(0) = \dfrac{1}{\sqrt{2}}

. Then the value of

y(\log_{e} 2)

is

(A) $\sqrt{\dfrac{5 + \sqrt{35}}{2}}$
(B) $\sqrt{\dfrac{7 + \sqrt{53}}{2}}$
(C) $\dfrac{7 + \sqrt{53}}{2}$
(D) $\dfrac{5 + \sqrt{35}}{2}$

Q2·MathematicsMultiple correctJEE Advanced 2026

Let

y = f(x)

be the real valued function defined on the interval

(0, \infty)

, satisfying

y(1) = 0

and the differential equation

x\dfrac{dy}{dx} = y - x^{3}

. Then which of the following statements is (are) TRUE ?

(A)The function $f$ has a local minimum at $x = \dfrac{1}{\sqrt{3}}$
(B)The function $f$ has a local maximum at $x = \dfrac{1}{\sqrt{3}}$
(C)The function $f$ is increasing in the interval $(1, 2)$
(D)If $g(x) = 4x^{3} - 5x^{2} + \dfrac{3}{2}x$ for $x > 0$ , then the number of elements in the set $\{x \in (0, \infty) : f(x) = g(x)\}$ is 2

Q3·MathematicsSingle correctJEE Main 2026

Let

y = y(x)

be the solution of the differential equation

x\sqrt{1-x^2}\,dy + \left(y\sqrt{1-x^2} - x\cos^{-1}x\right)dx = 0

,

x \in (0,1)

,

\displaystyle\lim_{x \to 1^-} y(x) = 1

. Then

y\left(\dfrac{1}{2}\right)

equals:

(A) $3 - \dfrac{\pi}{\sqrt{3}}$
(B) $4 - \sqrt{3}\pi$
(C) $4 - \dfrac{2\pi}{\sqrt{3}}$
(D) $3 - \dfrac{\pi}{2\sqrt{3}}$

Q4·MathematicsNumericalJEE Main 2026

Let y = y(x) be the solution of the differential equation

(x^2 - x\sqrt{x^2-1})dy + (y(x - \sqrt{x^2-1}) - x)dx = 0

, x ≥ 1. If y(1) = 1, then the greatest integer less than

y(\sqrt{5})

is ______.

Q5·MathematicsSingle correctJEE Main 2026

Let

f : \mathbb{R} \to \mathbb{R}

be such that

f(xy) = f(x)f(y)

, for all

x, y \in \mathbb{R}

and

f(0) \neq 0

. Let

g : [1, \infty) \to \mathbb{R}

be a differentiable function such that

x^{2}g(x) = \int_{1}^{x} (t^{2}f(t) - tg(t))\, dt.

Then

g(2)

is equal to :

(A) $\frac{13}{8}$
(B) $\frac{11}{16}$
(C) $\frac{15}{32}$
(D) $\frac{17}{64}$

Q6·MathematicsNumericalJEE Main 2026

Let

y = y(x)

be the solution of the differential equation

x\sin\left(\frac{y}{x}\right) dy = \left(y\sin\left(\frac{y}{x}\right) - x\right) dx

,

y(1) = \frac{\pi}{2}

and let

\alpha = \cos\left(\frac{y(e^{12})}{e^{12}}\right)

. Then the number of integral values of

p

, for which the equation

x^2 + y^2 - 2px + 2py + \alpha + 2 = 0

represents a circle of radius

r \leq 6

, is __________.

Q7·MathematicsNumericalJEE Main 2026

Let y = y(x) be the solution of the differential equation

(\tan x)^{1/2}\, dy = (\sec^{3} x - (\tan x)^{3/2} y)\, dx

,

0 < x < \frac{\pi}{2}

,

y\left(\frac{\pi}{4}\right) = \frac{6\sqrt{2}}{5}

. If

y\left(\frac{\pi}{3}\right) = \frac{4}{5}\alpha

, then

\alpha^{4}

equals _______.

Q8·MathematicsSingle correctJEE Main 2026

Let f : [1, ∞) → ℝ be a differentiable function defined as

f(x) = \int_{1}^{x} f(t)\, dt + (1 - x)(\log_e x - 1) + e

. Then the value of f(f(1)) is :

(A) $(1 + e^{e})$
(B)(1 + e)
(C) $(1 + e + e^{e})$
(D)1 + 2e

Q9·MathematicsSingle correctJEE Main 2026

Let

y = y(x)

be the solution of the differential equation:

\frac{dy}{dx} + \left(\frac{6x^{2} + (3x^{2} + 2x^{3} + 4)e^{-2x}}{(x^{3} + 2)(2 + e^{-2x})}\right)y = 2 + e^{-2x}

,

x \in (-1, 2)

, satisfying

y(0) = \frac{3}{2}

. If

y(1) = \alpha(2 + e^{-2})

, then α is equal to:

(A) $\frac{13}{8}$
(B) $\frac{6}{13}$
(C) $\frac{12}{13}$
(D) $\frac{13}{12}$

Q10·MathematicsSingle correctJEE Main 2026

Let

y = y(x)

be the solution of the differential equation

\frac{dy}{dx} = (1 + x + x^2)(1 - y + y^2)

,

y(0) = \frac{1}{2}

. Then

(2y(1) - 1)

is equal to:

(A) $\sqrt{3} \tan\left(\frac{11\sqrt{3}}{6}\right)$
(B) $\frac{\sqrt{3}}{2} \tan\left(\frac{11\sqrt{3}}{12}\right)$
(C) $\sqrt{3} \tan\left(\frac{11\sqrt{3}}{12}\right)$
(D) $\frac{\sqrt{3}}{2} \tan\left(\frac{11\sqrt{3}}{6}\right)$

Q11·MathematicsSingle correctJEE Main 2026

Let

x = x(y)

be the solution of the differential equation

2y^2 \frac{dx}{dy} - 2xy + x^2 = 0

,

y > 1

,

x(e) = e

. Then

x(e^2)

is equal to:

(A) $\frac{3}{2}e^2$
(B) $\frac{2}{3}e^2$
(C) $e^2$
(D) $2e^2$

Q12·MathematicsSingle correctJEE Main 2026

Let

y = y(x)

be the solution curve of the differential equation

(1 + \sin x)\frac{dy}{dx} + (y + 1)\cos x = 0

,

y(0) = 0

. If the curve

y = y(x)

passes through the point

\left(\alpha, \frac{-1}{2}\right)

, then a value of

\alpha

is :

(A) $\frac{\pi}{6}$
(B) $\frac{\pi}{4}$
(C) $\frac{\pi}{3}$
(D) $\frac{\pi}{2}$

Q13·MathematicsSingle correctJEE Main 2026

Let y = y(x) be the solution of the differential equation

x\frac{dy}{dx} - y = x^2 \cot x, x \in (0, \pi)

. If

y\left(\frac{\pi}{2}\right) = \frac{\pi}{2}

, then

6y\left(\frac{\pi}{6}\right) - 8y\left(\frac{\pi}{4}\right)

is equal to :

(A) $3\pi$
(B)- $3\pi$
(C)- $\pi$
(D) $\pi$

Q14·MathematicsSingle correctJEE Main 2026

Let

y = y(x)

be a differentiable function in the interval

(0, \infty)

such that

y(1) = 2

. and

\lim\limits_{t \to x} \left( \dfrac{t^{2} y(x) - x^{2} y(t)}{x - t} \right) = 3

for each

x > 0

. Then

2y(2)

is equal to

(A)18
(B)23
(C)27
(D)12

Q15·MathematicsNumericalJEE Main 2026

If the solution curve y = f(x) of the differential equation

(x^2 - 4)y' - 2xy + 2x(4 - x^2)^2 = 0

x > 2, passes through the point (3, 15), then the local maximum value of f is ………

Q16·MathematicsNumericalJEE Main 2026

Let f be a twice differentiable non-negative function such that

(f(x))^{2} = 25 + \int\limits_{0}^{x}\left((f(t))^{2} + (f'(t))^{2}\right) dt

. Then the mean of

f(\log_{e}(1))

,

f(\log_{e}(2))

, .....,

f(\log_{e}(625))

is equal to _________ .

Q17·MathematicsSingle correctJEE Main 2026

Let y = y(x) be the solution of the differential equation

x^4 dy + (4x^3 y + 2\sin x)dx = 0

,

x > 0

,

y\left(\frac{\pi}{2}\right) = 0

. Then

\pi^4 y\left(\frac{\pi}{3}\right)

is equal to :

(A)81
(B)92
(C)64
(D)72

Q18·MathematicsSingle correctJEE Main 2026

Let the solution curve of the differential equation

xdy - ydx = \sqrt{x^{2} + y^{2}}\,dx

,

x > 0

,

y(1) = 0

, be

y = y(x)

. Then

y(3)

is equal to

(A)4
(B)6
(C)1
(D)2

Q19·MathematicsSingle correctJEE Main 2026

Let

f : [1, \infty) \rightarrow \mathbb{R}

be a differentiable function, If

6\int_1^x f(t)\,dt = 3xf(x) + x^3 - 4

for all

x \ge 1

, then the value of f(2) − f(3) is

(A)−4
(B)−3
(C)4
(D)3

Q20·MathematicsSingle correctJEE Main 2026

Let

y = y(x)

be the solution of the differential equation

\sec x \frac{dy}{dx} - 2y = 2 + 3\sin x

,

x \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)

,

y(0) = -\frac{7}{4}

. Then

y\left(\frac{\pi}{6}\right)

is equal to:

(A) $-\frac{5}{2}$
(B) $-\frac{5}{4}$
(C) $-3\sqrt{3} - 7$
(D) $-3\sqrt{2} - 7$

Q21·MathematicsSingle correctJEE Main 2026

Let

y = y(x)

be the solution curve of the differential equation

(1 + x^{2})dy + (y - \tan^{-1}x)dx = 0

,

y(0) = 1

. Then the value of

y(1)

is :

(A) $\frac{2}{e^{\frac{\pi}{4}}} + \frac{\pi}{4} - 1$
(B) $\frac{2}{e^{\frac{\pi}{4}}} - \frac{\pi}{4} - 1$
(C) $\frac{4}{e^{\frac{\pi}{4}}} + \frac{\pi}{2} - 1$
(D) $\frac{4}{e^{\frac{\pi}{4}}} - \frac{\pi}{2} - 1$

Q22·MathematicsNumericalJEE Advanced 2025

Let y(x) be the solution of the differential equation

x^2 \frac{dy}{dx} + xy = x^2 + y^2,\, x > \frac{1}{e}

, satisfying y(1) = 0. Then the value of

2\frac{\left(y(e)\right)^2}{y\left(e^2\right)}

is ______

Q23·MathematicsNumericalJEE Advanced 2025

For all

x > 0

, let

y_1(x)

,

y_2(x)

, and

y_3(x)

be the functions satisfying

\frac{dy_1}{dx} - \left(\sin x\right)^2 y_1 = 0

,

y_1(1) = 5

,

\frac{dy_2}{dx} - \left(\cos x\right)^2 y_2 = 0

,

y_2(1) = \frac{1}{3}

,

\frac{dy_3}{dx} - \left(\frac{2 - x^3}{x^3}\right) y_3 = 0

,

y_3(1) = \frac{3}{5e}

, respectively. Then

\lim_{x \to 0^{+}} \frac{y_1(x) y_2(x) y_3(x) + 2x}{e^{3x} \sin x}

is equal to __________ .

Q24·MathematicsSingle correctJEE Main 2025

Let

f(x)=x-1

and

g(x)=e^x

for

x\in\mathbb{R}

. If

\dfrac{dy}{dx}=\left(e^{-2\sqrt x}\,g\big(f(f(x))\big)-\dfrac{y}{\sqrt x}\right)

,

y(0)=0

, then

y(1)

is:

(A) $\dfrac{1-e^2}{e^4}$
(B) $\dfrac{2e-1}{e^3}$
(C) $\dfrac{e-1}{e^4}$
(D) $\dfrac{1-e^3}{e^4}$

Q25·MathematicsSingle correctJEE Main 2025

Let

y=y(x)

be the solution curve of the differential equation

x(x^2+e^x)dy+(e^x(x-2)y-x^3)dx=0

,

x>0

, passing through the point

(1,0)

. Then

y(2)

is equal to:

(A) $\dfrac{4}{4-e^2}$
(B) $\dfrac{2}{2+e^2}$
(C) $\dfrac{2}{2-e^2}$
(D) $\dfrac{4}{4+e^2}$

Q26·MathematicsSingle correctJEE Main 2025

Let

y=y(x)

be the solution of the differential equation

(x^2+1)y'-2xy=(x^4+2x^2+1)\cos x

,

y(0)=1

. Then

\displaystyle\int_{-3}^{3} y(x)\,dx

is:

(A)24
(B)36
(C)30
(D)18

Q27·MathematicsSingle correctJEE Main 2025

If a curve

y=y(x)

passes through the point

\left(1,\dfrac{\pi}{2}\right)

and satisfies the differential equation

(7x^4\cot y-e^x\operatorname{cosec}y)\dfrac{dx}{dy}=x^5

,

x\ge1

, then at

x=2

, the value of

\cos y

is:

(A) $\dfrac{2e^2-e}{64}$
(B) $\dfrac{2e^2+e}{64}$
(C) $\dfrac{2e^2-e}{128}$
(D) $\dfrac{2e^2+e}{128}$

Q28·MathematicsSingle correctJEE Main 2025

Let

y=y(x)

be the solution of the differential equation

\dfrac{dy}{dx}+3(\tan^2 x)\,y+3y=\sec^2 x

,

y(0)=\dfrac{1}{3}+e^3

. Then

y\!\left(\dfrac{\pi}{4}\right)

is equal to:

(A) $\dfrac{2}{3}$
(B) $\dfrac{4}{3}$
(C) $\dfrac{4}{3}+e^3$
(D) $\dfrac{2}{3}+e^3$

Q29·MathematicsSingle correctJEE Main 2025

Let

g

be a differentiable function such that

\displaystyle\int_0^x g(t)\,dt=x-\int_0^x t\,g(t)\,dt

,

x\ge 0

and let

y=y(x)

satisfy the differential equation

\dfrac{dy}{dx}-y\tan x=2(x+1)\sec x\,g(x)

,

x\in\left[0,\dfrac{\pi}{2}\right)

. If

y(0)=0

, then

y\!\left(\dfrac{\pi}{3}\right)

is equal to:

(A) $\dfrac{2\pi}{3\sqrt3}$
(B) $\dfrac{4\pi}{3}$
(C) $\dfrac{2\pi}{3}$
(D) $\dfrac{4\pi}{3\sqrt3}$

Q30·MathematicsIntegerJEE Main 2025

Let

f:R\to R

be a thrice differentiable odd function satisfying

f''(x)\ge0

,

f''(x)=f(x)

,

f(0)=0

,

f'(0)=3

. Then

9f(\log_e 3)

is equal to ______.

Q31·MathematicsIntegerJEE Main 2025

Let

y=y(x)

be the solution of the differential equation

\dfrac{dy}{dx}+2y\sec^2x=2\sec^2x+3\tan x\cdot\sec^2x

such that

y(0)=\dfrac{5}{4}

. Then

12\left(y\left(\dfrac{\pi}{4}\right)-e^{-2}\right)

is equal to ______.

Q32·MathematicsSingle correctJEE Main 2025

Let

y=y(x)

be the solution of the differential equation

\cos x\,(\log_e(\cos x))^2\,dy+(\sin x-3y\sin x\,\log_e(\cos x))\,dx=0

,

x\in\left(0,\dfrac{\pi}{2}\right)

. If

y\left(\dfrac{\pi}{4}\right)=\dfrac{-1}{\log_e 2}

, then

y\left(\dfrac{\pi}{6}\right)

is:

(A) $\dfrac{2}{\log_e 3-\log_e 4}$
(B) $\dfrac{1}{\log_e 4-\log_e 3}$
(C) $\dfrac{-1}{\log_e 4}$
(D) $\dfrac{1}{\log_e 3-\log_e 4}$

Q33·MathematicsSingle correctJEE Main 2025

If for the solution curve

y=f(x)

of the differential equation

\dfrac{dy}{dx}+(\tan x)y=\dfrac{2+\sec x}{(1+2\sec x)^2}

,

x\in\left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right)

,

f\left(\dfrac{\pi}{3}\right)=\dfrac{\sqrt{3}}{10}

, then

f\left(\dfrac{\pi}{4}\right)

is equal to:

(A) $\dfrac{9\sqrt{3}+3}{10(4+\sqrt{3})}$
(B) $\dfrac{\sqrt{3}+1}{10(4+\sqrt{3})}$
(C) $\dfrac{5-\sqrt{3}}{2\sqrt{2}}$
(D) $\dfrac{4-\sqrt{2}}{14}$

Q34·MathematicsIntegerJEE Main 2025

If

y=y(x)

is the solution of the differential equation

\sqrt{4-x^2}\frac{dy}{dx}=\left(\left(\sin^{-1}\frac{x}{2}\right)^2-y\right)\sin^{-1}\frac{x}{2}

,

-2\le x\le2

,

y(2)=\frac{\pi^2-8}{4}

, then

y^2(0)

is equal to ______.

Q35·MathematicsSingle correctJEE Main 2025

Let for some function

y=f(x)

,

\int_0^x t\,f(t)dt=x^2 f(x)

,

x>0

and

f(2)=3

. Then

f(6)

is equal to:

(A)1
(B)2
(C)6
(D)3

Q36·MathematicsSingle correctJEE Main 2025

Let

y=y(x)

be the solution of the differential equation

(xy-5x^2\sqrt{1+x^2})dx+(1+x^2)dy=0

,

y(0)=0

. Then

y(\sqrt3)

is equal to

(A) $\dfrac{5\sqrt3}{2}$
(B) $\sqrt{\dfrac{14}{3}}$
(C) $2\sqrt2$
(D) $\sqrt{\dfrac{15}{2}}$

Q37·MathematicsSingle correctJEE Main 2025

Let

f:(0,\infty)\to\mathbb{R}

be a function differentiable at all points of its domain and satisfies the condition

x^2 f'(x)=2x f(x)+3

, with

f(1)=4

. Then

2f(2)

is equal to:

(A)29
(B)19
(C)39
(D)23

Q38·MathematicsIntegerJEE Main 2025

Let f be a differentiable function such that

2(x+2)^2 f(x)-3(x+2)^2=10\int_0^x (t+2)f(t)\,dt

,

x\ge 0

. Then

f(2)

is equal to _______

Q39·MathematicsIntegerJEE Main 2025

Let

y=y(x)

be the solution of the differential equation

2\cos x\dfrac{dy}{dx}=\sin 2x-4y\sin x

,

x\in\left(0,\dfrac{\pi}{2}\right)

. If

y\left(\dfrac{\pi}{3}\right)=0

, then

y\left(\dfrac{\pi}{4}\right)+y'\left(\dfrac{\pi}{4}\right)

is equal to __________.

Q40·MathematicsSingle correctJEE Main 2025

Let

x=x(y)

be the solution of the differential equation

y=\left(x-y\dfrac{dx}{dy}\right)\sin\!\left(\dfrac{x}{y}\right),\ y>0

and

x(1)=\dfrac{\pi}{2}

. Then

\cos\big(x(2)\big)

is equal to :

(A) $1-2(\log_e 2)^2$
(B) $2(\log_e 2)^2-1$
(C) $2(\log_e 2)-1$
(D) $1-2(\log_e 2)$

Q41·MathematicsSingle correctJEE Main 2025

Let a curve

y=f(x)

pass through the points

(0,5)

and

(\log_e 2, k)

. If the curve satisfies the differential equation

2(3+y)e^{2x}dx-(7+e^{2x})dy=0

, then

k

is equal to

(A) $16$
(B) $8$
(C) $32$
(D) $4$

Q42·MathematicsIntegerJEE Main 2025

Let

y=f(x)

be the solution of the differential equation

\dfrac{dy}{dx}+\dfrac{xy}{x^2-1}=\dfrac{x^6+4x}{\sqrt{1-x^2}}

,

-1<x<1

such that

f(0)=0

. If

6\displaystyle\int_{-1/2}^{1/2}f(x)\,dx=2\pi-\alpha

then

\alpha^2

is equal to ______.

Q43·MathematicsSingle correctJEE Main 2025

Let

x=x(y)

be the solution of

\dfrac{dx}{dy}+\dfrac{x}{y^{2}}=\dfrac{1}{y^{3}}

with

x(1)=1

. Then

x\big(\tfrac12\big)

is:

(A) $\tfrac12+e$
(B) $\tfrac32+e$
(C)3-e
(D)3+e

Q44·MathematicsSingle correctJEE Main 2025

If

x=f(y)

is the solution of the differential equation

(1+y^2)+\left(x-2e^{\tan^{-1}y}\right)\dfrac{dy}{dx}=0

,

y\in\left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right)

with

f(0)=1

, then

f\left(\dfrac{1}{\sqrt{3}}\right)

is equal to:

(A) $e^{\pi/4}$
(B) $e^{\pi/12}$
(C) $e^{\pi/3}$
(D) $e^{\pi/6}$

Q45·MathematicsSingle correctJEE Advanced 2024

Let

f(x)

be a continuously differentiable function on the interval

(0, \infty)

such that

f(1) = 2

and

\lim_{t \to x} \frac{t^{10} f(x) - x^{10} f(t)}{t^{9} - x^{9}} = 1

for each

x > 0

. Then, for all

x > 0

,

f(x)

is equal to

(A) $\frac{31}{11x} - \frac{9}{11} x^{10}$
(B) $\frac{9}{11x} + \frac{13}{11} x^{10}$
(C) $\frac{-9}{11x} + \frac{31}{11} x^{10}$
(D) $\frac{13}{11x} + \frac{9}{11} x^{10}$

Q46·MathematicsNumericalJEE Main 2024

For a differentiable function

f:\mathbb{R}\to\mathbb{R}

,

f'(x)=3f(x)+\alpha

, where

\alpha\in\mathbb{R}

,

f(0)=1

and

\displaystyle\lim_{x\to-\infty}f(x)=7

. Then

9f(-\log_{e}3)

is equal to _______.

Q47·MathematicsSingle correctJEE Main 2024

The solution curve, of the differential equation

2y\frac{dy}{dx} + 3 = 5\frac{dy}{dx}

, passing through the point

(0, 1)

is a conic, whose vertex lies on the line:

(A) $2x + 3y = 9$
(B) $2x + 3y = -9$
(C) $2x + 3y = -6$
(D) $2x + 3y = 6$

Q48·MathematicsSingle correctJEE Main 2024

The solution of the differential equation

(x^2 + y^2)dx - 5xy\, dy = 0

,

y(1) = 0

, is :

(A) $\left|x^2 - 4y^2\right|^5 = x^2$
(B) $\left|x^2 - 2y^2\right|^6 = x$
(C) $\left|x^2 - 4y^2\right|^6 = x$
(D) $\left|x^2 - 2y^2\right|^5 = x^2$

Q49·MathematicsSingle correctJEE Main 2024

Let

\displaystyle\int_{0}^{x}\sqrt{1-(y'(t))^{2}}\,dt=\int_{0}^{x}y(t)\,dt

,

0\le x\le 3

,

y\ge 0

,

y(0)=0

. Then at

x=2

,

y''+y+1

is equal to:

(A) $1$
(B) $2$
(C) $\sqrt{2}$
(D) $\dfrac{1}{2}$

Q50·MathematicsSingle correctJEE Main 2024

Let

y=y(x)

be the solution of the differential equation

(1+y^2)e^{\tan x}\,dx+\cos^2 x(1+e^{2\tan x})\,dy=0

,

y(0)=1

. Then

y\left(\dfrac\pi4\right)

is equal to:

(A) $\dfrac2e$
(B) $\dfrac{1}{e^2}$
(C) $\dfrac1e$
(D) $\dfrac{2}{e^2}$

Q51·MathematicsSingle correctJEE Main 2024

Let

y=y(x)

be the solution curve of the differential equation

\sec y\,\dfrac{dy}{dx}+2x\sin y=x^3\cos y

,

y(1)=0

. Then

y(\sqrt{3})

is equal to

(A) $\tfrac{\pi}{3}$
(B) $\tfrac{\pi}{6}$
(C) $\tfrac{\pi}{4}$
(D) $\tfrac{\pi}{12}$

Q52·MathematicsSingle correctJEE Main 2024

Let

f(x)

be a positive function such that the area bounded by

y=f(x)

,

y=0

from

x=0

to

x=a>0

is

e^{-a}+4a^2+a-1

. Then the differential equation, whose general solution is

y=c_1 f(x)+c_2

, where

c_1

and

c_2

are arbitrary constants, is:

(A) $(8e^x-1)\dfrac{d^2y}{dx^2}+\dfrac{dy}{dx}=0$
(B) $(8e^x-1)\dfrac{d^2y}{dx^2}-\dfrac{dy}{dx}=0$
(C) $(8e^x+1)\dfrac{d^2y}{dx^2}+\dfrac{dy}{dx}=0$
(D) $(8e^x+1)\dfrac{d^2y}{dx^2}-\dfrac{dy}{dx}=0$

Q53·MathematicsNumericalJEE Main 2024

Let

\alpha|x|=|y|e^{xy-\beta}

,

\alpha,\beta\in\mathbb{N}

be the solution of the differential equation

x\,dy-y\,dx+xy(x\,dy+y\,dx)=0

,

y(1)=2

. Then

\alpha+\beta

is equal to _______ .

Q54·MathematicsSingle correctJEE Main 2024

Let

y=y(x)

be the solution of the differential equation

(1+x^{2})\dfrac{dy}{dx}+y=e^{\tan^{-1}x}

,

y(1)=0

. Then

y(0)

is

(A) $\tfrac{1}{4}\left(e^{\pi/2}-1\right)$
(B) $\tfrac{1}{2}\left(1-e^{\pi/2}\right)$
(C) $\tfrac{1}{4}\left(1-e^{\pi/2}\right)$
(D) $\tfrac{1}{2}\left(e^{\pi/2}-1\right)$

Q55·MathematicsSingle correctJEE Main 2024

Suppose the solution of the differential equation

\frac{dy}{dx} = \frac{(2+\alpha)x - \beta y + 2}{\beta x - 2\alpha y - (\beta\gamma - 4\alpha)}

represents a circle passing through origin. Then the radius of this circle is:

(A) $\sqrt{17}$
(B) $\tfrac{1}{2}$
(C) $\tfrac{\sqrt{17}}{2}$
(D)2

Q56·MathematicsNumericalJEE Main 2024

If the solution

y(x)

of the given differential equation

(e^{y}+1)\cos x\,dx+e^{y}\sin x\,dy=0

passes through the point

\left(\frac{\pi}{2},0\right)

, then the value of

e^{y\left(\frac{\pi}{6}\right)}

is equal to ______.

Q57·MathematicsSingle correctJEE Main 2024

Let

y=y(x)

be the solution of the differential equation

(2x\log_{e}x)\dfrac{dy}{dx}+2y=\dfrac{3}{x}\log_{e}x

,

x>0

and

y(e^{-1})=0

. Then

y(e)

is equal to

(A) $\tfrac{3}{2e}$
(B) $\tfrac{2}{3e}$
(C) $-\tfrac{3}{e}$
(D) $\tfrac{2}{e}$

Q58·MathematicsNumericalJEE Main 2024

Let

f

be a differentiable function in the interval

(0, \infty)

such that

f(1) = 1

and

\displaystyle\lim_{t \to x} \dfrac{t^2 f(x) - x^2 f(t)}{t - x} = 1

for each

x > 0

. Then

2f(2) + 3f(3)

is equal to ___.

Q59·MathematicsSingle correctJEE Main 2024

The differential equation of the family of circles passing through the origin and having their centre on the line

y=x

is:

(A) $(x^2-y^2+2xy)\,dx=(x^2-y^2+2xy)\,dy$
(B) $(x^2+y^2+2xy)\,dx=(x^2+y^2-2xy)\,dy$
(C) $(x^2-y^2+2xy)\,dx=(x^2-y^2-2xy)\,dy$
(D) $(x^2+y^2-2xy)\,dx=(x^2+y^2+2xy)\,dy$

Q60·MathematicsSingle correctJEE Main 2024

If

y = y(x)

is the solution of the differential equation

\dfrac{dy}{dx} + 2y = \sin(2x)

,

y(0) = \dfrac{3}{4}

, then

y\left(\dfrac{\pi}{8}\right)

is equal to:

(A) $e^{-\pi/8}$
(B) $e^{-\pi/4}$
(C) $e^{\pi/4}$
(D) $e^{\pi/8}$

Q61·MathematicsNumericalJEE Main 2024

Let

y=y(x)

be the solution of the differential equation

\dfrac{dy}{dx}+\dfrac{2x}{(1+x^2)^2}\,y=x\,e^{\frac{1}{1+x^2}}

;

y(0)=0

. Then the area enclosed by the curve

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

and the line

y-x=4

is __________.

Q62·MathematicsNumericalJEE Main 2024

Let

y=y(x)

be the solution of the differential equation

(x+y+2)^2\,dx=dy

,

y(0)=-2

. Let the maximum and minimum values of the function

y=y(x)

in

\left[0,\dfrac{\pi}{3}\right]

be

\alpha

and

\beta

, respectively. If

(3\alpha+\pi)^2+\beta^2=\gamma+\delta\sqrt{3}

,

\gamma,\delta\in\mathbb{Z}

, then

\gamma+\delta

equals

Q63·MathematicsSingle correctJEE Main 2024

If the solution

y=y(x)

of the differential equation

(x^4+2x^3+3x^2+2x+2)\,dy=(2x^2+2x+3)\,dx

satisfies

y(-1)=-\dfrac\pi4

, then

y(0)

is equal to:

(A) $-\dfrac{\pi}{12}$
(B) $0$
(C) $\dfrac\pi4$
(D) $\dfrac\pi2$

Q64·MathematicsSingle correctJEE Main 2024

Let

y=y(x)

be the solution of the differential equation

(x^2+4)^2\,dy+(2x^3y+8xy-2)\,dx=0

. If

y(0)=0

, then

y(2)

is equal to

(A) $\tfrac{\pi}{8}$
(B) $\tfrac{\pi}{16}$
(C) $2\pi$
(D) $\tfrac{\pi}{32}$

Q65·MathematicsNumericalJEE Main 2024

Let the solution

y=y(x)

of the differential equation

\dfrac{dy}{dx}-y=1+4\sin x

satisfy

y(\pi)=1

. Then

y\left(\dfrac\pi2\right)+10

is equal to ___

Q66·MathematicsNumericalJEE Main 2024

If

\dfrac{dx}{dy}=\dfrac{1+x-y^2}{y}

,

x(1)=1

, then

5x(2)

is equal to __________.

Q67·MathematicsSingle correctJEE Main 2024

Let

\alpha

be a non-zero real number. Suppose

f:\mathbb R\to\mathbb R

is a differentiable function such that

f(0)=2

and

\displaystyle\lim_{x\to-\infty}f(x)=1

. If

f'(x)=\alpha f(x)+3

, for all

x\in\mathbb R

, then

f(-\log_e 2)

is equal to:

(A)3
(B)5
(C)9
(D)7

Q68·MathematicsNumericalJEE Main 2024

If

y=x(t)

is the solution of the differential equation

(t+1)\,dx=\left(2x+(t+1)^4\right)dt

,

x(0)=2

, then

x(1)

equals ___

Q69·MathematicsSingle correctJEE Main 2024

Let

y=y(x)

be the solution of the differential equation

\dfrac{dy}{dx}=2x(x+y)^3-x(x+y)-1

,

y(0)=1

. Then

\left(\dfrac{1}{\sqrt2}+y\left(\dfrac{1}{\sqrt2}\right)\right)^2

equals:

(A) $\dfrac{1}{4+\sqrt e}$
(B) $\dfrac{3}{3-\sqrt e}$
(C) $\dfrac{2}{1+\sqrt e}$
(D) $\dfrac{1}{2-\sqrt e}$

Q70·MathematicsSingle correctJEE Main 2024

Let

y=y(x)

be the solution of the differential equation

\dfrac{dy}{dx}=\dfrac{\tan x+y}{\sin x(\sec x-\sin x\tan x)}

,

x\in\left(0,\dfrac{\pi}{2}\right)

satisfying the condition

y\left(\dfrac{\pi}{4}\right)=2

. Then,

y\left(\dfrac{\pi}{3}\right)

is

(A) $\sqrt{3}\left(2+\log_e\sqrt{3}\right)$
(B) $\dfrac{\sqrt{3}}{2}\left(2+\log_e 3\right)$
(C) $\sqrt{3}\left(1+2\log_e 3\right)$
(D) $\sqrt{3}\left(2+\log_e 3\right)$

Q71·MathematicsSingle correctJEE Main 2024

The temperature

T(t)

of a body at time

t=0

is

160^\circ

F and it decreases continuously as per the differential equation

\dfrac{dT}{dt}=-K(T-80)

, where K is a positive constant. If

T(15)=120^\circ

F, then

T(45)

is equal to

(A) $85^\circ$ F
(B) $95^\circ$ F
(C) $90^\circ$ F
(D) $80^\circ$ F

Q72·MathematicsSingle correctJEE Main 2024

The solution curve of the differential equation

y\dfrac{dx}{dy}=x\left(\log_e x-\log_e y+1\right)

,

x>0,\ y>0

passing through the point

(e,1)

is

(A) $\left|\log_e\dfrac{y}{x}\right|=x$
(B) $\left|\log_e\dfrac{y}{x}\right|=y^2$
(C) $\left|\log_e\dfrac{x}{y}\right|=y$
(D) $2\left|\log_e\dfrac{x}{y}\right|=y+1$

Q73·MathematicsNumericalJEE Main 2024

Let

y=y(x)

be the solution of the differential equation

\sec^2 x\,dx+\left(e^{2y}\tan^2 x+\tan x\right)dy=0

,

0<x<\dfrac{\pi}{2}

,

y\left(\dfrac{\pi}{4}\right)=0

. If

y\left(\dfrac{\pi}{6}\right)=\alpha

, Then

e^{8\alpha}

is equal to ______.

Q74·MathematicsNumericalJEE Main 2024

Let

Y=Y(X)

be a curve lying in the first quadrant such that the area enclosed by the line

Y-y=Y'(x)(X-x)

and the co-ordinate axes, where

(x,y)

is any point on the curve, is always

\dfrac{-y^2}{2Y'(x)}+1

,

Y'(x)\ne 0

. If

Y(1)=1

, then

12\,Y(2)

equals ______.

Q75·MathematicsSingle correctJEE Main 2024

Let

y=y(x)

be the solution of the differential equation

\sec x\,dy+\{2(1-x)\tan x+x(2-x)\}dx=0

such that

y(0)=2

. Then

y(2)

is equal to:

(A) $2$
(B) $2(1-\sin(2))$
(C) $2\sin(2)+1$
(D) $1$

Q76·MathematicsNumericalJEE Main 2024

Let

y=y(x)

be the solution of the differential equation

(1-x^2)\,dy=\left[xy+(x^3+2)\sqrt{3(1-x^2)}\right]dx

,

-1<x<1

,

y(0)=0

. If

y\left(\dfrac12\right)=\dfrac{m}{n}

,

m

and

n

are co-prime numbers, then

m+n

is equal to ___

Q77·MathematicsSingle correctJEE Main 2024

If

\sin\left(\dfrac{y}{x}\right)=\log_e|x|+\dfrac{\alpha}{2}

is the solution of the differential equation

x\cos\left(\dfrac{y}{x}\right)\dfrac{dy}{dx}=y\cos\left(\dfrac{y}{x}\right)+x

and

y(1)=\dfrac{\pi}{3}

, then

\alpha^2

is equal to:

(A)3
(B)12
(C)4
(D)9

Q78·MathematicsNumericalJEE Main 2024

If the solution curve

y=y(x)

of the differential equation

\left(1+y^2\right)\left(1+\log_e x\right)dx+x\,dy=0

,

x>0

passes through the point

(1,1)

and

y(e)=\dfrac{\alpha-\tan\left(\frac{3}{2}\right)}{\beta+\tan\left(\frac{3}{2}\right)}

, then

\alpha+2\beta

is ______.

Q79·MathematicsSingle correctJEE Main 2024

A function

y=f(x)

satisfies

f(x)\sin 2x+\sin x-(1+\cos^2 x)f'(x)=0

with condition

f(0)=0

. Then

f\left(\dfrac{\pi}{2}\right)

is equal to

(A) $1$
(B) $0$
(C) $-1$
(D) $2$

Q80·MathematicsNumericalJEE Main 2024

If the solution curve, of the differential equation

\dfrac{dy}{dx}=\dfrac{x+y-2}{x-y}

passing through the point

(2,1)

is

\tan^{-1}\left(\dfrac{y-1}{x-1}\right)-\dfrac{1}{\beta}\log_e\left(\alpha+\left(\dfrac{y-1}{x-1}\right)^2\right)=\log_e|x-1|

, then

5\beta+\alpha

is equal to __________.

Q81·MathematicsSingle correctJEE Main 2024

Let

x=x(t)

and

y=y(t)

be solutions of the differential equations

\dfrac{dx}{dt}+ax=0

and

\dfrac{dy}{dt}+by=0

respectively,

a,b\in\mathbb R

. Given that

x(0)=2

,

y(0)=1

and

3y(1)=2x(1)

, the value of

t

, for which

x(t)=y(t)

, is:

(A) $\log_{2/3} 2$
(B) $\log_4 3$
(C) $\log_3 4$
(D) $\log_{4/3} 2$

Q82·MathematicsNumericalJEE Main 2024

If the solution of the differential equation

(2x+3y-2)\,dx+(4x+6y-7)\,dy=0

,

y(0)=3

, is

\alpha x+\beta y+3\log_e|2x+3y-\gamma|=6

, then

\alpha+2\beta+3\gamma

is equal to __________.

Q83·MathematicsNumericalJEE Main 2024

Let for a differentiable function

f:(0,\infty)\to\mathbb R

,

f(x)-f(y)\ge\log_e(x/y)+x-y

,

\forall\,x,y\in(0,\infty)

. Then

\displaystyle\sum_{n=1}^{20}f'\!\left(\dfrac{1}{n^2}\right)

is equal to __________.

Q84·MathematicsSingle correctJEE Main 2024

If

y=y(x)

is the solution curve of the differential equation

(x^2-4)\,dy-(y^2-3y)\,dx=0

,

x>2

,

y(4)=\dfrac32

and the slope of the curve is never zero, then the value of

y(10)

equals :

(A) $\dfrac{3}{1+(8)^{1/4}}$
(B) $\dfrac{3}{1+2\sqrt2}$
(C) $\dfrac{3}{1-2\sqrt2}$
(D) $\dfrac{3}{1-(8)^{1/4}}$

Q85·MathematicsIntegerJEE Advanced 2023

For

x \in \mathbb{R}

, let

y(x)

be a solution of the differential equation

(x^{2} - 5)\frac{dy}{dx} - 2xy = -2x(x^{2} - 5)^{2}

such that

y(2) = 7

. Then the maximum value of the function

y(x)

is

Q86·MathematicsSingle correctJEE Advanced 2023

Let

f : [1, \infty) \to \mathbb{R}

be a differentiable function such that

f(1) = \frac{1}{3}

and

3\int_{1}^{x} f(t)\,dt = x f(x) - \frac{x^{3}}{3}

,

x \in [1, \infty)

. Let e denote the base of the natural logarithm. Then the value of

f(e)

is

(A) $\frac{e^{2}+4}{3}$
(B) $\frac{\log_{e} 4 + e}{3}$
(C) $\frac{4e^{2}}{3}$
(D) $\frac{e^{2}-4}{3}$

Q87·MathematicsSingle correctJEE Main 2023

Let

x=x(y)

be the solution of the differential equation

2(y+2)\log_e(y+2)\,dx+(x+4-2\log_e(y+2))\,dy=0

,

y>-1

with

x(e^4-2)=1

. Then

x(e^9-2)

is equal to

(A) $\frac{4}{9}$
(B) $\frac{10}{3}$
(C) $\frac{32}{9}$
(D)3

Q88·MathematicsSingle correctJEE Main 2023

Let

y=y_{1}(x)

and

y=y_{2}(x)

be the solution curves of the differential equation

\dfrac{dy}{dx}=y+7

with initial conditions

y_{1}(0)=0,\,y_{2}(0)=1

respectively. Then the curves

y=y_{1}(x)

and

y=y_{2}(x)

intersect at:

(A)Two points
(B)no point
(C)infinite number of points
(D)one point

Q89·MathematicsNumericalJEE Main 2023

If

y = y(x)

is the solution of the differential equation

\dfrac{dy}{dx} + \dfrac{4x}{(x^2-1)}y = \dfrac{x+2}{(x^2-1)^{\frac{5}{2}}}

,

x > 1

such that

y(2) = \dfrac{2}{9}\log_e\left(2 + \sqrt{3}\right)

and

y\left(\sqrt{2}\right) = \alpha\log_e\left(\sqrt{\alpha} + \beta\right) + \beta - \sqrt{\gamma}

,

\alpha, \beta, \gamma \in \mathbb{N}

, then

\alpha\beta\gamma

is equal to

Q90·MathematicsSingle correctJEE Main 2023

Let

y=y(x)

,

y>0

, be a solution curve of the differential equation

(1+x^2)\,dy=y(x-y)\,dx

. If

y(0)=1

and

y(2\sqrt2)=\beta

, then

(A) $e^{1/\beta}=e(3+2\sqrt2)$
(B) $e^{1/\beta}=e^{-2}(5+4\sqrt2)$
(C) $e^{1/\beta}=e^{-2}(3+2\sqrt2)$
(D) $e^{1/\beta}=e(5+4\sqrt2)$

Q91·MathematicsSingle correctJEE Main 2023

Let

y = y(x)

be a solution curve of the differential equation,

(1 - x^2 y^2) dx = y\, dx + x\, dy

. If the line

x = 1

intersects the curve

y = y(x)

at

y = 2

and the line

x = 2

intersects the curve

y = y(x)

at

y = \alpha

, then a value of

\alpha

is

(A) $\frac{3e^2}{2(3e^2 - 1)}$
(B) $\frac{3e^2}{2(3e^2 + 1)}$
(C) $\frac{1 - 3e^2}{2(3e^2 + 1)}$
(D) $\frac{1 + 3e^2}{2(3e^2 - 1)}$

Q92·MathematicsSingle correctJEE Main 2023

Let

y=y(x)

be the solution of the differential equation

\dfrac{dy}{dx}+\dfrac{5}{x\left(x^5+1\right)}y=\dfrac{\left(x^5+1\right)^2}{x^7}

,

x>0

. If

y(1)=2

, then

y(2)

is equal to

(A) $\dfrac{637}{128}$
(B) $\dfrac{679}{128}$
(C) $\dfrac{693}{128}$
(D) $\dfrac{697}{128}$

Q93·MathematicsSingle correctJEE Main 2023

The slope of tangent at any point

(x,y)

on a curve

y=y(x)

is

\dfrac{x^2+y^2}{2xy}

,

x>0

. If

y(2)=0

, then a value of

y(8)

is

(A) $-2\sqrt3$
(B) $4\sqrt3$
(C) $2\sqrt3$
(D) $-4\sqrt2$

Q94·MathematicsNumericalJEE Main 2023

Let the tangent at any point P on a curve passing through the points

(1,1)

and

\left(\frac{1}{10},100\right)

, intersect the positive x-axis and y-axis at the points A and B respectively. If

PA:PB=1:k

and

y=y(x)

is the solution of the differential equation

e^{\frac{dy}{dx}}=kx+\frac{k}{2}

,

y(0)=k

, then

4y(1)-5\log_e 3

is equal to _______ .

Q95·MathematicsNumericalJEE Main 2023

If the solution curve of the differential equation

(y-2\log_{e}x)dx+(x\log_{e}x^{2})dy=0

,

x>1

passes through the points

\left(e,\frac{4}{3}\right)

and

(e^{4},\alpha)

, then

\alpha

is equal to

Q96·MathematicsNumericalJEE Main 2023

Let the solution curve

x=x(y)

,

0<y<\dfrac{\pi}{2}

, of the differential equation

(\log_e(\cos y))^{2}\cos y\,dx-(1+3x\log_e(\cos y))\sin y\,dy=0

satisfy

x\left(\dfrac{\pi}{3}\right)=\dfrac{1}{2\log_e 2}

. If

x\left(\dfrac{\pi}{6}\right)=\dfrac{1}{\log_e m-\log_e n}

, where m and n are co-prime, then mn is equal to

Q97·MathematicsNumericalJEE Main 2023

Let a curve

y=f(x)

,

x\in(0,\infty)

pass through the points

P\left(1,\frac{3}{2}\right)

and

Q\left(a,\frac{1}{2}\right)

. If the tangent at any point

R(b,f(b))

to the given curve cuts the y-axis at the point

S(0,c)

such that

bc=3

, then

(PQ)^2

is equal to

Q98·MathematicsSingle correctJEE Main 2023

If the solution curve

f(x, y) = 0

of the differential equation

(1+\log_e x)\frac{dx}{dy}-x\log_e x=e^y

,

x>0

, passes through the points

(1,0)

and

(\alpha,2)

then

\alpha^\alpha

is equal to

(A) $e^{2e^{\sqrt{2}}}$
(B) $e^{\sqrt{2}e^2}$
(C) $e^{e^2}$
(D) $e^{2e^2}$

Q99·MathematicsNumericalJEE Main 2023

Let

y=y(x)

be a solution of the differential equation

(x\cos x)dy+(xy\sin x+y\cos x-1)dx=0,\,0<x<\dfrac{\pi}{2}

. If

y\left(\dfrac{\pi}{3}\right)=\dfrac{\sqrt3}{\pi}

, then

\left|\dfrac{\pi}{6}\,y\left(\dfrac{\pi}{6}\right)+2y\left(\dfrac{\pi}{6}\right)\right|

is equal to _____.

Q100·MathematicsSingle correctJEE Main 2023

If

y=y(x)

is the solution curve of the differential equation

\frac{dy}{dx}+y\tan x=x\sec x

,

0\le x\le\frac{\pi}{3}

,

y(0)=1

, then

y\!\left(\frac{\pi}{6}\right)

is equal to

(A) $\frac{\pi}{12}-\frac{\sqrt{3}}{2}\log_e\!\left(\frac{2\sqrt{3}}{e}\right)$
(B) $\frac{\pi}{12}-\frac{\sqrt{3}}{2}\log_e\!\left(\frac{2}{e\sqrt{3}}\right)$
(C) $\frac{\pi}{12}+\frac{\sqrt{3}}{2}\log_e\!\left(\frac{2}{e\sqrt{3}}\right)$
(D) $\frac{\pi}{12}+\frac{\sqrt{3}}{2}\log_e\!\left(\frac{2\sqrt{3}}{e}\right)$

Q101·MathematicsSingle correctJEE Main 2023

Let

\alpha x=\exp(x^{\beta}y^{\gamma})

be the solution of the differential equation

2x^2y\,dy-(1-xy^2)dx=0

,

x>0

,

y(2)=\sqrt{\log_e 2}

. Then

\alpha+\beta+\gamma

equals:

(A) $1$
(B) $-1$
(C) $3$
(D) $0$

Q102·MathematicsSingle correctJEE Main 2023

The area enclosed by the closed curve

C

given by the differential equation

\frac{dy}{dx}+\frac{x+a}{y-2}=0

,

y(1)=0

is

4\pi

. Let

P

and

Q

be the points of intersection of the curve

C

and the

y

-axis. If normals at

P

and

Q

on the curve

C

intersect

x

-axis at points

R

and

S

respectively, then the length of the line segment

RS

is

(A) $2$
(B) $\frac{4\sqrt{3}}{3}$
(C) $2\sqrt{3}$
(D) $\frac{2\sqrt{3}}{3}$

Q103·MathematicsNumericalJEE Main 2023

Let

f:\mathbb{R}\to\mathbb{R}

be a differentiable function such that

f'(x)+f(x)=\int_0^2 f(t)\,dt

. If

f(0)=e^{-2}

, then

2f(0)-f(2)

is equal to _______ .

Q104·MathematicsSingle correctJEE Main 2023

Let

y=y(x)

be the solution of the differential equation

(3y^2-5x^2)y\,dx+2x(x^2-y^2)dy=0

such that

y(1)=1

. Then

\left|(y(2))^3-12y(2)\right|

is equal to:

(A) $16\sqrt2$
(B) $32\sqrt2$
(C) $32$
(D) $64$

Q105·MathematicsSingle correctJEE Main 2023

Let a differentiable function

f

satisfy

f(x)+\int_{3}^{x}\frac{f(t)}{t}\,dt=\sqrt{x+1},\ x\ge 3

. Then

12f(8)

is equal to :

(A)34
(B)1
(C)17
(D)19

Q106·MathematicsSingle correctJEE Main 2023

Let the solution curve

y=y(x)

of the differential equation

\dfrac{dy}{dx}-\dfrac{3x^5\tan^{-1}(x^3)}{(1+x^6)^{3/2}}y=2x\exp\left\{\dfrac{x^3-\tan^{-1}x^3}{\sqrt{1+x^6}}\right\}

pass through the origin. Then

y(1)

is equal to:

(A) $\exp\left(\dfrac{4+\pi}{4\sqrt2}\right)$
(B) $\exp\left(\dfrac{1-\pi}{4\sqrt2}\right)$
(C) $\exp\left(\dfrac{\pi-4}{4\sqrt2}\right)$
(D) $\exp\left(\dfrac{4-\pi}{4\sqrt2}\right)$

Q107·MathematicsSingle correctJEE Main 2023

The solution of the differential equation

\dfrac{dy}{dx}=-\left(\dfrac{x^2 + 3y^2}{3x^2 + y^2}\right)

,

y(1)=0

, is:

(A) $\log_e|x + y| - \dfrac{xy}{(x + y)^2}=0$
(B) $\log_e|x + y| + \dfrac{2xy}{(x + y)^2}=0$
(C) $\log_e|x + y| - \dfrac{2xy}{(x + y)^2}=0$
(D) $\log_e|x + y| + \dfrac{xy}{(x + y)^2}=0$

Q108·MathematicsSingle correctJEE Main 2023

Let

y=y(x)

be the solution of the differential equation

x\log_e x\dfrac{dy}{dx}+y=x^2\log_e x,\,(x>1)

. If

y(2)=2

, then

y(e)

is equal to:

(A) $\dfrac{1+e^2}{2}$
(B) $\dfrac{4+e^2}{4}$
(C) $\dfrac{2+e^2}{2}$
(D) $\dfrac{1+e^2}{4}$

Q109·MathematicsSingle correctJEE Main 2023

Let

y = f(x)

be the solution of the differential equation

y(x + 1)dx - x^2 dy = 0

,

y(1) = e

. Then

\lim_{x \to 0^+} f(x)

is equal to

(A) $\dfrac{1}{e^2}$
(B) $e^2$
(C)0
(D) $\dfrac{1}{e}$

Q110·MathematicsSingle correctJEE Main 2023

Let

y=y(x)

be the solution curve of the differential equation

\dfrac{dy}{dx}=\dfrac{y}{x}\big(1+xy^2(1+\log_e x)\big)

,

x>0

,

y(1)=3

. Then

\dfrac{y^2(x)}{9}

is equal to:

(A) $\dfrac{x^2}{2x^3(2+\log_e x^3)-3}$
(B) $\dfrac{x^2}{3x^3(1+\log_e x^2)-2}$
(C) $\dfrac{x^2}{7-3x^3(2+\log_e x^2)}$
(D) $\dfrac{x^2}{5-2x^3(2+\log_e x^3)}$

Q111·MathematicsSingle correctJEE Main 2023

Let

y = y(t)

be a solution of the differential equation

\dfrac{dy}{dt} + \alpha y = \gamma e^{-\beta t}

where

\alpha > 0

,

\beta > 0

and

\gamma > 0

. Then

\displaystyle\lim_{t \to \infty} y(t)

(A)is $-1$
(B)is 1
(C)does not exist
(D)is 0

Q112·MathematicsSingle correctJEE Main 2023

Let

y=y(x)

be the solution of the differential equation

x^3\,dy+(xy-1)\,dx=0

,

x>0

,

y\!\left(\tfrac{1}{2}\right)=3-e

. Then

y(1)

is equal to

(A) $1$
(B) $e$
(C) $3$
(D) $2-e$

Q113·MathematicsSingle correctJEE Main 2023

Let

y=y(x)

be the solution of the differential equation

(x^2-3y^2)dx+3xy\,dy=0,\ y(1)=1

. Then

6y^2(e)

is equal to

(A) $2e^2$
(B) $3e^2$
(C) $e^2$
(D) $\dfrac32 e^2$

Q114·MathematicsNumericalJEE Main 2023

Let

f

be a differentiable function defined on

\left[0,\dfrac{\pi}{2}\right]

such that

f(x)>0

and

f(x)+\int_0^x f(t)\sqrt{1-(\log_e f(t))^2}\,dt=e,\ \forall x\in\left[0,\dfrac{\pi}{2}\right]

. Then

\left(6\log_e f\left(\dfrac{\pi}{6}\right)\right)^2

is equal to

Q115·MathematicsIntegerJEE Advanced 2022

If y(x) is the solution of the differential equation

xdy - \left(y^{2} - 4y\right)dx = 0

for x > 0, y(1) = 2, and the slope of the curve y = y(x) is never zero, then the value of

10y\left(\sqrt{2}\right)

is ________.

Q116·MathematicsMultiple correctJEE Advanced 2022

For x ∈ R, let the function y(x) be the solution of the differential equation

\frac{dy}{dx} + 12y = \cos\left(\frac{\pi}{12}x\right)

, y(0) = 0 . Then, which of the following statements is/are TRUE?

(A)y(x) is an increasing function
(B)y(x) is a decreasing function
(C)There exists a real number β such that the line y = β intersects the curve y = y(x) at infinitely many points
(D)y(x) is a periodic function

Q117·MathematicsSingle correctJEE Main 2022

Let

y = y(x)

be the solution curve of the differential equation

\frac{dy}{dx} + \left( \frac{2x^2 + 11x + 13}{x^3 + 6x^2 + 11x + 6} \right) y = \frac{(x + 3)}{x + 1}

,

x > -1

, which passes through the point

(0, 1)

. Then

y(1)

is equal to:

(A) $\frac{1}{2}$
(B) $\frac{3}{2}$
(C) $\frac{5}{2}$
(D) $\frac{7}{2}$

Q118·MathematicsSingle correctJEE Main 2022

Let the solution curve

y = y(x)

of the differential equation

(1 + e^{2x})\left(\frac{dy}{dx} + y\right) = 1

pass through the point

\left(0, \frac{\pi}{2}\right)

. Then,

\lim_{x \to \infty} e^{x} y(x)

is equal to :

(A) $\frac{\pi}{4}$
(B) $\frac{3\pi}{4}$
(C) $\frac{\pi}{2}$
(D) $\frac{3\pi}{2}$

Q119·MathematicsSingle correctJEE Main 2022

If the solution curve of the differential equation

\frac{dy}{dx} = \frac{x + y - 2}{x - y}

passes through the point

(2, 1)

and

(k + 1, 2)

,

k > 0

, then

(A) $2\tan^{-1}\left( \frac{1}{k} \right) = \log_e \left( k^2 + 1 \right)$
(B) $\tan^{-1}\left( \frac{1}{k} \right) = \log_e \left( k^2 + 1 \right)$
(C) $2\tan^{-1}\left( \frac{1}{k + 1} \right) = \log_e \left( k^2 + 2k + 2 \right)$
(D) $2\tan^{-1}\left( \frac{1}{k} \right) = \log_e \left( \frac{k^2 + 1}{k^2} \right)$

Q120·MathematicsSingle correctJEE Main 2022

Let the solution curve of the differential equation

xdy = \left(\sqrt{x^{2}+y^{2}} + y\right)dx

,

x > 0

, intersect the line

x = 1

at

y = 0

and the line

x = 2

at

y = \alpha

. Then the value of

\alpha

is :

(A) $\frac{1}{2}$
(B) $\frac{3}{2}$
(C) $-\frac{3}{2}$
(D) $\frac{5}{2}$

Q121·MathematicsSingle correctJEE Main 2022

If

y = y(x)

,

x \in \left(0, \frac{\pi}{2}\right)

be the solution curve of the differential equation

\left(\sin^{2}2x\right)\frac{dy}{dx} + \left(8\sin^{2}2x + 2\sin 4x\right)y = 2e^{-4x}\left(2\sin 2x + \cos 2x\right)

, with

y\left(\frac{\pi}{4}\right) = e^{-\pi}

, then

y\left(\frac{\pi}{6}\right)

is equal to :

(A) $\frac{2}{\sqrt{3}}e^{-2\pi/3}$
(B) $\frac{2}{\sqrt{3}}e^{2\pi/3}$
(C) $\frac{1}{\sqrt{3}}e^{-2\pi/3}$
(D) $\frac{1}{\sqrt{3}}e^{2\pi/3}$

Q122·MathematicsSingle correctJEE Main 2022

The differential equation of the family of circles passing through the points

(0,2)

and

(0,-2)

is

(A) $2xy\frac{dy}{dx}+\left(x^{2}-y^{2}+4\right)=0$
(B) $2xy\frac{dy}{dx}+\left(x^{2}+y^{2}-4\right)=0$
(C) $2xy\frac{dy}{dx}+\left(y^{2}-x^{2}+4\right)=0$
(D) $2xy\frac{dy}{dx}-\left(x^{2}-y^{2}+4\right)=0$

Q123·MathematicsSingle correctJEE Main 2022

Let

y=y(x)

be the solution curve of the differential equation

\frac{dy}{dx}+\frac{1}{x^{2}-1}y=\left(\frac{x-1}{x+1}\right)^{\frac{1}{2}}

,

x>1

passing through the point

\left(2,\sqrt{\frac{1}{3}}\right)

. Then

\sqrt{7}y(8)

is equal to

(A) $11+6\log_{e}3$
(B)19
(C) $12-2\log_{e}3$
(D) $19-6\log_{e}3$

Q124·MathematicsSingle correctJEE Main 2022

Consider a curve

y = y(x)

in the first quadrant as shown in the figure. Let the area

A_{1}

is twice the area

A_{2}

. Then the normal to the curve perpendicular to the line

2x - 12y = 15

does NOT pass through the point.

(A) $(6, 21)$
(B) $(8, 9)$
(C) $(10, -4)$
(D) $(12, -15)$

Q125·MathematicsSingle correctJEE Main 2022

Let

y = y_1(x)

and

y = y_2(x)

be two distinct solutions of the differential equation

\frac{dy}{dx} = x + y

, with

y_1(0) = 0

and

y_2(0) = 1

respectively. Then, the number of points of intersection of

y = y_1(x)

and

y = y_2(x)

is

(A)0
(B)1
(C)2
(D)3

Q126·MathematicsNumericalJEE Main 2022

Let y = y(x) be the solution curve of the differential equation

\sin\left(2x^{2}\right)\log_{e}\left(\tan x^{2}\right)dy+\left(4xy-4\sqrt{2}x\sin\left(x^{2}-\frac{\pi}{4}\right)\right)dx=0

,

0<x<\sqrt{\frac{\pi}{2}}

, which passes through the point

\left(\sqrt{\frac{\pi}{6}},1\right)

. Then

\left|y\left(\sqrt{\frac{\pi}{3}}\right)\right|

is equal to ______.

Q127·MathematicsNumericalJEE Main 2022

Let a curve

y = y\left(x\right)

pass through the point (3, 3) and the area of the region under this curve, above the x-axis and between the abscissae 3 and x(>3) be

\left(\frac{y}{x}\right)^{3}

. If this curve also passes through the point

\left(\alpha, 6\sqrt{10}\right)

in the first quadrant, then α is equal to _______

Q128·MathematicsNumericalJEE Main 2022

Suppose

y=y(x)

be the solution curve to the differential equation

\frac{dy}{dx}-y=2-e^{-x}

such that

\lim_{x\to\infty}y(x)

is finite. If a and b are respectively the x- and y- intercepts of the tangent to the curve at

x=0

, then the value of

a-4b

is equal to __________.

Q129·MathematicsSingle correctJEE Main 2022

Let the solution curve

y = f(x)

of the differential equation

\frac{dy}{dx}+\frac{xy}{x^{2}-1}=\frac{x^{4}+2x}{\sqrt{1-x^{2}}}, x\in\left(-1,1\right)

pass through the origin. Then

\int\limits_{-\frac{\sqrt{3}}{2}}^{\frac{\sqrt{3}}{2}}f\left(x\right)dx

is equal to

(A) $\frac{\pi}{3}-\frac{1}{4}$
(B) $\frac{\pi}{3}-\frac{\sqrt{3}}{4}$
(C) $\frac{\pi}{6}-\frac{\sqrt{3}}{4}$
(D) $\frac{\pi}{6}-\frac{\sqrt{3}}{2}$

Q130·MathematicsSingle correctJEE Main 2022

If

\dfrac{dy}{dx}

+ 2y tan x = sin x, 0 < x <

\dfrac{\pi}{2}

and

y\left(\dfrac{\pi}{3}\right)

= 0, then the maximum value of y(x) is

(A) $\dfrac{1}{8}$
(B) $\dfrac{3}{4}$
(C) $\dfrac{1}{4}$
(D) $\dfrac{3}{8}$

Q131·MathematicsSingle correctJEE Main 2022

The slope of the tangent to a curve C : y = y(x) at any point

[x, y)

on it is

\frac{2e^{2x} - 6e^{-x} + 9}{2 + 9e^{-2x}}

. If C passes through the points

\left( 0, \frac{1}{2} + \frac{\pi}{2\sqrt{2}} \right)

and

\left( \alpha, \frac{1}{2} e^{2\alpha} \right)

then

e^{\alpha}

is equal to :

(A) $\frac{3 + \sqrt{2}}{3 - \sqrt{2}}$
(B) $\frac{3}{\sqrt{2}} \left( \frac{3 + \sqrt{2}}{3 - \sqrt{2}} \right)$
(C) $\frac{1}{\sqrt{2}} \left( \frac{\sqrt{2} + 1}{\sqrt{2} - 1} \right)$
(D) $\frac{\sqrt{2} + 1}{\sqrt{2} - 1}$

Q132·MathematicsSingle correctJEE Main 2022

The general solution of the differential equation

(x - y^{2})dx + y(5x + y^{2})dy = 0

is :

(A) $(y^{2} + x)^{4} = C|(y^{2} + 2x)^{3}|$
(B) $(y^{2} + 2x)^{4} = C|(y^{2} + x)^{3}|$
(C) $|(y^{2} + x)^{3}| = C(2y^{2} + x)^{4}$
(D) $|(y^{2} + 2x)^{3}| = C(2y^{2} + x)^{4}$

Q133·MathematicsSingle correctJEE Main 2022

Let the solution curve of the differential equation

x\frac{dy}{dx} - y = \sqrt{y^{2} + 16x^{2}}

,

y(1) = 3

be

y = y(x)

. Then

y(2)

is equal to :

(A)15
(B)11
(C)13
(D)17

Q134·MathematicsSingle correctJEE Main 2022

If

y = y(x)

is the solution of the differential equation

\left(1 + e^{2x}\right)\frac{dy}{dx} + 2\left(1 + y^{2}\right)e^{x} = 0

and

y(0) = 0

, then

6\left(y'(0) + \left(y\left(\log_{e}\sqrt{3}\right)\right)^{2}\right)

is equal to:

(A) $2$
(B) $-2$
(C) $-4$
(D) $-1$

Q135·MathematicsNumericalJEE Main 2022

Let

y = y(x)

,

x > 1

, be the solution of the differential equation

(x - 1)\frac{dy}{dx} + 2xy = \frac{1}{x - 1}

, with

y(2) = \frac{1 + e^4}{2e^4}

. If

y(3) = \frac{e^\alpha + 1}{\beta e^\alpha}

. then the value of

\alpha + \beta

is equal to______.

Q136·MathematicsNumericalJEE Main 2022

Let y = y(x) be the solution of the differential equation

\frac{dy}{dx} + \frac{\sqrt{2}\,y}{2\cos^4 x - \cos 2x} = x e^{\tan^{-1}(\sqrt{2}\cot 2x)}, 0 < x < \pi/2

with

y\left(\frac{\pi}{4}\right) = \frac{\pi^2}{32}

. If

y\left(\frac{\pi}{3}\right) = \frac{\pi^2}{18} e^{-\tan^{-1}(\alpha)}

, then the value of

3\alpha^2

is equal to _____.

Q137·MathematicsSingle correctJEE Main 2022

Let

x = x(y)

be the solution of the differential equation

2y\,e^{x/y^{2}}dx + \left(y^{2} - 4xe^{x/y^{2}}\right)dy = 0

such that

x(1) = 0

. Then,

x(e)

is equal to

(A) $e\log_{e}(2)$
(B) $-e\log_{e}(2)$
(C) $e^{2}\log_{e}(2)$
(D) $-e^{2}\log_{e}(2)$

Q138·MathematicsSingle correctJEE Main 2022

Let

y = y(x)

be the solution of the differential equation

x\left(1-x^{2}\right)\frac{dy}{dx}+\left(3x^{2}y-y-4x^{3}\right)=0, x>1,

with

y(2) = -2.

Then

y(3)

is equal to

(A) $-18$
(B) $-12$
(C) $-6$
(D) $-3$

Q139·MathematicsSingle correctJEE Main 2022

Let the slope of the tangent to a curve

y = f(x)

at

(x, y)

be given by

2\tan x(\cos x - y)

. if the curve passes through the point

\left(\frac{\pi}{4}, 0\right)

, then the value of

\int_{0}^{\pi/2} y\,dx

is equal to

(A) $\left(2 - \sqrt{2}\right) + \frac{\pi}{\sqrt{2}}$
(B) $2 - \frac{\pi}{\sqrt{2}}$
(C) $\left(2 + \sqrt{2}\right) + \frac{\pi}{\sqrt{2}}$
(D) $2 + \frac{\pi}{\sqrt{2}}$

Q140·MathematicsSingle correctJEE Main 2022

Let the solution curve

y = y(x)

of the differential equation,

\left[\frac{x}{\sqrt{x^{2}-y^{2}}}+e^{\frac{y}{x}}\right]x\frac{dy}{dx}=x+\left[\frac{x}{\sqrt{x^{2}-y^{2}}}+e^{\frac{y}{x}}\right]y

pass through the points (1, 0) and

(2\alpha, \alpha), \alpha > 0.

Then

\alpha

is equal to

(A) $\frac{1}{2}\exp\left(\frac{\pi}{6}+\sqrt{e}-1\right)$
(B) $\frac{1}{2}\exp\left(\frac{\pi}{3}+\sqrt{e}-1\right)$
(C) $\exp\left(\frac{\pi}{6}+\sqrt{e}+1\right)$
(D) $2\exp\left(\frac{\pi}{3}+\sqrt{e}-1\right)$

Q141·MathematicsNumericalJEE Main 2022

Let v be the solution of the differential equation

(1-x^{2})dy =\left(xy+(x^{3}+2)\sqrt{1-x^{2}}\right)dx,\ -1<x<1

and y (0) = 0 if

\int_{-\frac{1}{2}}^{\frac{1}{2}}\sqrt{1-x^{2}}\;y(x)dx=k

then

k^{-1}

is equal to :

Q142·MathematicsSingle correctJEE Main 2022

Let

\frac{dy}{dx} = \frac{ax - by + a}{bx + cy + a}

, where a, b, c are constants, represent a circle passing through the point (2, 5). Then the shortest distance of the point (11, 6) from this circle is :

(A)10
(B)8
(C)7
(D)5

Q143·MathematicsSingle correctJEE Main 2022

If the solution curve of the differential equation

\left((\tan^{-1}y)-x\right)dy=(1+y^{2})dx

passes through the point (1, 0) then the abscissa of the point on the curve whose ordinate is

\tan(1)

is :

(A) $2e$
(B) $\dfrac{2}{e}$
(C) $2$
(D) $\dfrac{1}{e}$

Q144·MathematicsNumericalJEE Main 2022

Let

S = (0, 2\pi) - \left\{\frac{\pi}{2}, \frac{3\pi}{4}, \frac{3\pi}{2}, \frac{7\pi}{4}\right\}

. Let

y = y(x)

,

x \in S

, be the solution curve of the differential equation

\frac{dy}{dx} = \frac{1}{1 + \sin 2x}

,

y\left(\frac{\pi}{4}\right) = \frac{1}{2}

. if the sum of abscissas of all the points of intersection of the curve

y = y(x)

with the curve

y = \sqrt{2}\sin x

is

\frac{k\pi}{12}

, then k is equal to __________.

Q145·MathematicsSingle correctJEE Main 2022

If y = y(x) is the solution of the differential equation

x\frac{dy}{dx}+2y=xe^x,\ y(1)=0

, then the local maximum value of the function

z(x)=x^2y(x)-e^x

, x∈R is :

(A)1 – e
(B)0
(C) $\frac{1}{2}$
(D) $\frac{4}{e}-e$

Q146·MathematicsSingle correctJEE Main 2022

If the solution of the differential equation

\frac{dy}{dx}+e^x\left(x^2-2\right)y=\left(x^2-2x\right)\left(x^2-2\right)e^{2x}

satisfies y(0) = 0, then the value of y(2) is ________ .

(A)–1
(B)1
(C)0
(D)e

Q147·MathematicsNumericalJEE Main 2022

Let the solution curve

y = y(x)

of the differential equation

(4 + x^{2})dy - 2x(x^{2} + 3y + 4)dx = 0

pass through the origin. Then

y(2)

is equal to_________.

Q148·MathematicsSingle correctJEE Main 2022

If

y = y(x)

is the solution of the differential equation

2x^2 \frac{dy}{dx} - 2xy + 3y^2 = 0

such that

y(e) = \frac{e}{3}

, then

y(1)

is equal to

(A) $\frac{1}{3}$
(B) $\frac{2}{3}$
(C) $\frac{3}{2}$
(D)3

Q149·MathematicsSingle correctJEE Main 2022

Let

y = y(x)

be the solution of the differential equation

(x + 1)y' - y = e^{3x}(x + 1)^2

, with

y(0) = \frac{1}{3}

. Then, the point

x = -\frac{4}{3}

for the curve

y = y(x)

is:

(A)not a critical point
(B)a point of local minima
(C)a point of local maxima
(D)a point of inflection

Q150·MathematicsSingle correctJEE Main 2022

If the solution curve

y = y(x)

of the differential equation

y^2dx + (x^2 - xy + y^2)dy = 0

, which passes through the point (1, 1) and intersects the line

y = \sqrt{3}\,x

at the point

(\alpha, \sqrt{3}\,\alpha)

, then value of

\log_e(\sqrt{3}\,\alpha)

is equal to

(A) $\frac{\pi}{3}$
(B) $\frac{\pi}{2}$
(C) $\frac{\pi}{12}$
(D) $\frac{\pi}{6}$

Q151·MathematicsSingle correctJEE Main 2022

The slope of normal at any point

(x, y)

,

x > 0

,

y > 0

on the curve

y = y(x)

is given by

\dfrac{x^2}{xy - x^2 y^2 - 1}

. If the curve passes through the point

(1, 1)

, then

e \cdot y(e)

is equal to

(A) $\dfrac{1-\tan(1)}{1+\tan(1)}$
(B) $\tan(1)$
(C) $1$
(D) $\dfrac{1+\tan(1)}{1-\tan(1)}$

Q152·MathematicsSingle correctJEE Main 2022

If

x=x(y)

is the solution of the differential equation

y\dfrac{dx}{dy}=2x+y^{3}(y+1)e^{y}

,

x(1)=0

; then

x(e)

is equal to :

(A) $e^{3}(e^{e}-1)$
(B) $e^{e}(e^{3}-1)$
(C) $e^{2}(e^{e}+1)$
(D) $e^{e}(e^{2}-1)$

Q153·MathematicsSingle correctJEE Main 2022

A particle is moving in the xy-plane along a curve C passing through the point (3, 3). The tangent to the curve C at the point P meets the x-axis at Q. If the y-axis bisects the segment PQ, then C is a parabola with

(A)length of latus rectum 3
(B)length of latus rectum 6
(C)focus $\left(\dfrac{4}{3},0\right)$
(D)focus $\left(0,\dfrac{3}{4}\right)$

Q154·MathematicsMultiple correctJEE Advanced 2021

For any real numbers α and β, let

y_{\alpha,\beta}(x)

, x ∈ ℝ, be the solution of the differential equation

\frac{dy}{dx} + \alpha y = xe^{\beta x}

,

y(1) = 1

Let

S = \{y_{\alpha,\beta}(x) : \alpha, \beta \in \mathbb{R}\}

. Then which of the following functions belong(s) to the set S?

(A) $f(x) = \frac{x^2}{2}e^{-x} + \left(e - \frac{1}{2}\right)e^{-x}$
(B) $f(x) = -\frac{x^2}{2}e^{-x} + \left(e + \frac{1}{2}\right)e^{-x}$
(C) $f(x) = \frac{e^x}{2}\left(x - \frac{1}{2}\right) + \left(e - \frac{e^2}{4}\right)e^{-x}$
(D) $f(x) = \frac{e^x}{2}\left(\frac{1}{2} - x\right) + \left(e + \frac{e^2}{4}\right)e^{-x}$

Q155·MathematicsSingle correctJEE Main 2021

If y = y (x) is the solution curve of the differential equation

x^{2}dy + \left(y - \frac{1}{x}\right)dx = 0

; x > 0 and y(1) = 1, then

y\left(\frac{1}{2}\right)

is equal to :

(A) $\frac{3}{2} - \frac{1}{\sqrt{e}}$
(B) $3 + \frac{1}{\sqrt{e}}$
(C)3 + e
(D)3 − e

Q156·MathematicsSingle correctJEE Main 2021

If

\frac{dy}{dx} = \frac{2^{x+y} - 2^{x}}{2^{y}}

,

y(0) = 1

, then

y(1)

is equal to :

(A) $\log_{2}(2 + e)$
(B) $\log_{2}(1 + e)$
(C) $\log_{2}(2e)$
(D) $\log_{2}(1 + e^{2})$

Q157·MathematicsSingle correctJEE Main 2021

If

y \frac{dy}{dx} = x\left[\frac{y^2}{x^2} + \frac{\phi\left(\frac{y^2}{x^2}\right)}{\phi'\left(\frac{y^2}{x^2}\right)}\right]

,

x > 0

,

\phi > 0

, and

y(1) = -1

, then

\phi\left(\frac{y^2}{4}\right)

is equal to :

(A) $4\,\phi(2)$
(B) $4\,\phi(1)$
(C) $2\,\phi(1)$
(D) $\phi(1)$

Q158·MathematicsSingle correctJEE Main 2021

If

\frac{dy}{dx} = \frac{2^x y + 2^y \cdot 2^x}{2^x + 2^{x+y} \log_e 2}

,

y(0) = 0

, then for

y = 1

, the value of

x

lies in the interval:

(A) $(1, 2)$
(B) $\left(\frac{1}{2}, 1\right]$
(C) $(2, 3)$
(D) $\left(0, \frac{1}{2}\right]$

Q159·MathematicsSingle correctJEE Main 2021

Let us consider a curve,

y = f(x)

passing through the point

(-2, 2)

and the slope of the tangent to the curve at any point

(x, f(x))

is given by

f(x) + xf'(x) = x^2

. Then :

(A) $x^2 + 2xf(x) - 12 = 0$
(B) $x^3 + xf(x) + 12 = 0$
(C) $x^3 - 3xf(x) - 4 = 0$
(D) $x^2 + 2xf(x) + 4 = 0$

Q160·MathematicsSingle correctJEE Main 2021

If the solution curve of the differential equation

(2x - 10y^{3})\,dy + y\,dx = 0

, passes through the points (0, 1) and (2, β), then β is a root of the equation:

(A) $y^{5} - 2y - 2 = 0$
(B) $2y^{5} - 2y - 1 = 0$
(C) $2y^{5} - y^{2} - 2 = 0$
(D) $y^{5} - y^{2} - 1 = 0$

Q161·MathematicsSingle correctJEE Main 2021

Let

y = y(x)

be the solution of the differential equation

\frac{dy}{dx} = 2(y + 2\sin x - 5)\, x - 2\cos x

such that

y(0) = 7

. Then

y(\pi)

is equal to :

(A) $2e^{\pi^2} + 5$
(B) $e^{\pi^2} + 5$
(C) $3e^{\pi^2} + 5$
(D) $7e^{\pi^2} + 5$

Q162·MathematicsSingle correctJEE Main 2021

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) form the line 3x + 4y = 5, is given by :

(A) $10\frac{d^{2}y}{dx^{2}} = 11$
(B) $11\frac{d^{2}x}{dy^{2}} = 10$
(C) $10\frac{d^{2}x}{dy^{2}} = 11$
(D) $11\frac{d^{2}y}{dx^{2}} = 10$

Q163·MathematicsSingle correctJEE Main 2021

Let

y = y(x)

be a solution curve of the differential equation

(y + 1)\tan^2 x\, dx + \tan x\, dy + y\, dx = 0

,

x \in \left(0, \frac{\pi}{2}\right)

. If

\lim_{x \to 0+} xy(x) = 1

, then the value of

y\left(\frac{\pi}{4}\right)

is :

(A) $-\frac{\pi}{4}$
(B) $\frac{\pi}{4} - 1$
(C) $\frac{\pi}{4} + 1$
(D) $\frac{\pi}{4}$

Q164·MathematicsSingle correctJEE Main 2021

Let

y(x)

be the solution of the differential equation

2x^{2}dy + (e^{y} - 2x)dx = 0

,

x > 0

. If

y(e) = 1

, then

y(1)

is equal to :

(A) $0$
(B) $2$
(C) $\log_{e} 2$
(D) $\log_{e}(2e)$

Q165·MathematicsNumericalJEE Main 2021

If

y=y\left(x\right), y \in \left[0,\frac{\pi}{2}\right]

is the solution of the differential equation

\sec y\frac{dy}{dx}-\sin\left(x+y\right)-\sin\left(x-y\right)=0

, with

y\left(0\right)=0

, then

5y'\left(\frac{\pi}{2}\right)

is equal to ____.

Q166·MathematicsSingle correctJEE Main 2021

Let

y = y\left(x\right)

be solution of the differential equation

\log_e\left(\frac{dy}{dx}\right) = 3x + 4y

, with

y\left(0\right) = 0

. If

y\left(-\frac{2}{3}\log_e 2\right) = \alpha \log_e 2

, then the value of

\alpha

is equal to :

(A) $\frac{1}{4}$
(B) $-\frac{1}{4}$
(C)2
(D) $-\frac{1}{2}$

Q167·MathematicsSingle correctJEE Main 2021

Let y = y(x) be the solution of the differential equation

\left( x - x^3 \right) dy = \left( y + yx^2 - 3x^4 \right) dx

, x > 2. If y(3) = 3 then y(4) is equal to

(A)12
(B)4
(C)8
(D)16

Q168·MathematicsNumericalJEE Main 2021

Let y = y

\left(x\right)

be the solution of the differential equation dy =

e^{\alpha x+y}

dx; α ∈ N . If y

\left(\log_{e}2\right)

=

\log_{e}2

and y(0) =

\log_{e}\left(\frac{1}{2}\right)

, then the value of α is equal to………….

Q169·MathematicsSingle correctJEE Main 2021

Let

y=y(x)

be the solution of the differential equation

\frac{dy}{dx}=1+xe^{y-x}

,

-\sqrt{2}<x<\sqrt{2}

,

y(0)=0

then, the minimum value of

y(x)

,

x \in \left(-\sqrt{2},\sqrt{2}\right)

is equal to :

(A) $\left(1+\sqrt{3}\right)-\log_{e}\left(\sqrt{3}-1\right)$
(B) $\left(2+\sqrt{3}\right)+\log_{e}2$
(C) $\left(1-\sqrt{3}\right)-\log_{e}\left(\sqrt{3}-1\right)$
(D) $\left(2-\sqrt{3}\right)-\log_{e}2$

Q170·MathematicsNumericalJEE Main 2021

Let

y = y(x)

be solution of the following differential equation

e^y \frac{dy}{dx} - 2e^y \sin x + \sin x \cos^2 x = 0

,

y\left( \frac{\pi}{2} \right) = 0

. If

y(0) = \log_e \left( \alpha + \beta e^{-2} \right)

, then

4\left( \alpha + \beta \right)

is equal to.......

Q171·MathematicsSingle correctJEE Main 2021

Let

y = y\left(x\right)

be the solution of the differential equation

xdy = \left(y + x^3\cos x\right)dx

with

y\left(\pi\right) = 0

, then

y\left(\frac{\pi}{2}\right)

is equal to :

(A) $\frac{\pi^2}{4} - \frac{\pi}{2}$
(B) $\frac{\pi^2}{4} + \frac{\pi}{2}$
(C) $\frac{\pi^2}{2} - \frac{\pi}{4}$
(D) $\frac{\pi^2}{2} + \frac{\pi}{4}$

Q172·MathematicsNumericalJEE Main 2021

Let a curve

y = f(x)

pass through the point

\left(2, \left(\log_{e}2\right)^{2}\right)

and have slope

\dfrac{2y}{x\log_{e}x}

for all positive real value of x. Then the value of

f(e)

is equal to..........

Q173·MathematicsNumericalJEE Main 2021

Let

y = y(x)

be the solution of the differential equation

\left((x + 2)e^{\frac{y+1}{x+2}} + (y + 1)\right)dx = (x + 2)dy

,

y(1) = 1

. If the domain

y = y(x)

is an open interval

(\alpha, \beta)

, then

|\alpha + \beta|

is equal to..........

Q174·MathematicsSingle correctJEE Main 2021

Let

y = y(x)

be the solution of the differential equation

\cosec^2 x \, dy + 2 \, dx = (1 + y \cos 2x) \cosec^2 x \, dx

, with

y\left(\dfrac{\pi}{4}\right) = 0

. Then, the value of

\left(y(0) + 1\right)^2

is equal to :

(A) $e^{1/2}$
(B) $e^{-1/2}$
(C) $e^{-1}$
(D) $e$

Q175·MathematicsSingle correctJEE Main 2021

Let

y=y(x)

satisfies the equation

\frac{dy}{dx}-|A|=0,

for all

x>0,

where

A=\begin{bmatrix} y & \sin x & 1 \\ 0 & -1 & 1 \\ 2 & 0 & \frac{1}{x} \end{bmatrix}

. If

y(\pi)=\pi+2,

then the value of

y\left(\frac{\pi}{2}\right)

is :

(A) $\frac{\pi}{2}+\frac{4}{\pi}$
(B) $\frac{3\pi}{2}-\frac{1}{\pi}$
(C) $\frac{\pi}{2}-\frac{4}{\pi}$
(D) $\frac{\pi}{2}-\frac{1}{\pi}$

Q176·MathematicsSingle correctJEE Main 2021

Let

y = y(x)

be the solution of the differential equation

x\tan\left(\frac{y}{x}\right)dy = \left(y\tan\left(\frac{y}{x}\right)-x\right)dx

,

-1 \leq x \leq 1

,

y\left(\frac{1}{2}\right) = \frac{\pi}{6}

Then the area of the region bounded by the curves

x = 0

,

x = \frac{1}{\sqrt{2}}

and

y = y(x)

in the upper half plane is :

(A) $\frac{1}{6}(\pi-1)$
(B) $\frac{1}{8}(\pi-1)$
(C) $\frac{1}{4}(\pi-2)$
(D) $\frac{1}{12}(\pi-3)$

Q177·MathematicsNumericalJEE Main 2021

Let a curve

y=y(x)

be given by the solution of the differential equation

\cos\left(\frac{1}{2}\cos^{-1}\left(e^{-x}\right)\right)dx=\sqrt{e^{2x}-1}\,dy

If it intersects y-axis at

y=-1

, and the intersection point of the curve with x-axis is

(\alpha,0)

, then

e^{\alpha}

is equal to……..

Q178·MathematicsSingle correctJEE Main 2021

Let

y = y(x)

be the solution of the differential equation

e^x\sqrt{1-y^2}\,dx+\left(\frac{y}{x}\right)dy = 0

,

y\left(1\right) = -1

Then the value of

\left(y(3)\right)^2

is equal to :

(A) $1+4e^6$
(B) $1-4e^3$
(C) $1+4e^3$
(D) $1-4e^6$

Q179·MathematicsSingle correctJEE Main 2021

The differential equation satisfied by the system of parabolas

y^{2} = 4a(x + a)

is :

(A) $y\left(\dfrac{dy}{dx}\right)^{2} - 2x\left(\dfrac{dy}{dx}\right) - y = 0$
(B) $y\left(\dfrac{dy}{dx}\right)^{2} - 2x\left(\dfrac{dy}{dx}\right) + y = 0$
(C) $y\left(\dfrac{dy}{dx}\right)^{2} + 2x\left(\dfrac{dy}{dx}\right) - y = 0$
(D) $y\left(\dfrac{dy}{dx}\right) + 2x\left(\dfrac{dy}{dx}\right) - y = 0$

Q180·MathematicsNumericalJEE Main 2021

Let y = y(x) be the solution of the differential equation

xdy - ydx = \sqrt{\left(x^{2} - y^{2}\right)}\,dx

, x ≥ 1, with y(1) = 0. If the area bounded by the line x = 1,

x = e^{\pi}

, y = 0 and y = y(x) is

\alpha e^{2\pi} + \beta

, then the value of 10(α + β) is equal to ________ .

Q181·MathematicsSingle correctJEE Main 2021

Let y = y(x) be the solution of the differential equation

\frac{dy}{dx} = (y+1)\left((y+1)e^{x^{2}/2} - x\right)

, 0 < x < 2.1, with y(2) = 0. Then the value of

\frac{dy}{dx}

at x = 1 is equal to :

(A) $\frac{-e^{3/2}}{\left(e^{2}+1\right)^{2}}$
(B) $-\frac{2e^{2}}{\left(1+e^{2}\right)^{2}}$
(C) $\frac{e^{5/2}}{\left(1+e^{2}\right)^{2}}$
(D) $\frac{5e^{1/2}}{\left(e^{2}+1\right)^{2}}$

Q182·MathematicsSingle correctJEE Main 2021

Let

y = y(x)

be the solution of the differential equation

\cos x\,(3\sin x + \cos x + 3)dy = (1 + y\sin x\,(3\sin x + \cos x + 3))dx

,

0 \le x \le \frac{\pi}{2}

,

y(0) = 0

. Then,

y\left(\frac{\pi}{3}\right)

is equal to:

(A) $2\log_{e}\left(\frac{2\sqrt{3}+9}{6}\right)$
(B) $2\log_{e}\left(\frac{2\sqrt{3}+10}{11}\right)$
(C) $2\log_{e}\left(\frac{\sqrt{3}+7}{2}\right)$
(D) $2\log_{e}\left(\frac{3\sqrt{3}-8}{4}\right)$

Q183·MathematicsSingle correctJEE Main 2021

If the curve

y = y(x)

is the solution of the differential equation

2(x^{2} + x^{5/4})dy - y(x + x^{1/4})dx = 2x^{9/4}dx

,

x > 0

which passes through the point

\left(1, 1-\frac{4}{3}\log_{e}2\right)

, then the value of

y(16)

is equal to :

(A) $4\left(\frac{31}{3}+\frac{8}{3}\log_{e}3\right)$
(B) $\left(\frac{31}{3}+\frac{8}{3}\log_{e}3\right)$
(C) $4\left(\frac{31}{3}-\frac{8}{3}\log_{e}3\right)$
(D) $\left(\frac{31}{3}-\frac{8}{3}\log_{e}3\right)$

Q184·MathematicsSingle correctJEE Main 2021

Which of the following is true for

y(x)

that satisfies the differential equation

\frac{dy}{dx} = xy - 1 + x - y

;

y(0) = 0

:

(A) $y(1) = e^{-\frac{1}{2}} - 1$
(B) $y(1) = e^{\frac{1}{2}} - e^{-\frac{1}{2}}$
(C) $y(1) = 1$
(D) $y(1) = e^{\frac{1}{2}} - 1$

Q185·MathematicsSingle correctJEE Main 2021

If

y = y(x)

is the solution of the differential equation

\frac{dy}{dx}+(\tan x)\,y=\sin x,\ 0 \le x \le \frac{\pi}{3}

, with

y(0) = 0

, then

y\left(\frac{\pi}{4}\right)

equal to :

(A) $\frac{1}{4}\log_{e}2$
(B) $\left(\frac{1}{2\sqrt{2}}\right)\log_{e}2$
(C) $\log_{e}2$
(D) $\frac{1}{2}\log_{e}2$

Q186·MathematicsSingle correctJEE Main 2021

Let

C_{1}

be the curve obtained by the solution of differential equation

2xy\frac{dy}{dx}=y^{2}-x^{2}

,

x > 0

. Let the curve

C_{2}

be the solution of

\frac{2xy}{x^{2}-y^{2}}=\frac{dy}{dx}

. If both the curves pass through (1,1), then the area enclosed by the curves

C_{1}

and

C_{2}

is equal to :

(A) $\pi-1$
(B) $\frac{\pi}{2}-1$
(C) $\pi+1$
(D) $\frac{\pi}{4}+1$

Q187·MathematicsSingle correctJEE Main 2021

If y = y(x) is the solution of the differential equation,

\frac{dy}{dx} + 2y\tan x = \sin x

,

y\left(\frac{\pi}{3}\right) = 0

, then the maximum value of the function y(x) over ℝ is equal to :

(A)8
(B) $\frac{1}{2}$
(C) $-\frac{15}{4}$
(D) $\frac{1}{8}$

Q188·MathematicsNumericalJEE Main 2021

Let the normals at all the points on a given curve pass through a fixed point (a, b). If the curve passes through (3, -3) and

\left(4, -2\sqrt{2}\right)

, and given that

a - 2\sqrt{2}\ b = 3

, then

(a^{2}+b^{2}+ab)

is equal to________________.

Q189·MathematicsSingle correctJEE Main 2021

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% in 2 hours. If the population of bacteria is 2000 after

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

hours, then

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

is equal to

(A)4
(B)2
(C)16
(D)8

Q190·MathematicsNumericalJEE Main 2021

The difference betweeen degree and order of differential equation that represents the family of curves given by

y^{2}=a\left(x+\frac{\sqrt{a}}{2}\right)

, a > 0 is _______.

Q191·MathematicsSingle correctJEE Main 2021

Let

f(x) = \int_{0}^{x} e^{t} f(t)dt + e^{x}

be a differentiable function for all

x \in R

. Then f(x) equals.

(A) $2e^{(e^{x}-1)} - 1$
(B) $e^{(e^{x}-1)}$
(C) $2e^{e^{x}} - 1$
(D) $e^{e^{x}} - 1$

Q192·MathematicsSingle correctJEE Main 2021

Let slope of the tangent line to a curve at any point P(x, y) be given by

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

. If the curve intersects the line x + 2y = 4 at x = − 2, then the value of y, for which the point (3, y) lies on the curve, is :

(A) $-\frac{18}{11}$
(B) $-\frac{18}{19}$
(C) $-\frac{4}{3}$
(D) $\frac{18}{35}$

Q193·MathematicsNumericalJEE Main 2021

If y = y(x) is the solution of the equation

e^{\sin y}\cos y\,\frac{dy}{dx}+e^{\sin y}\cos x=\cos x

, y(0) = 0 ; then

1+y\left(\frac{\pi}{6}\right)+\frac{\sqrt{3}}{2}y\left(\frac{\pi}{3}\right)+\frac{1}{\sqrt{2}}y\left(\frac{\pi}{4}\right)

is equal to _______.

Q194·MathematicsSingle correctJEE Main 2021

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

(x, y)

is

\dfrac{x^2 - 4x + y + 8}{x - 2}

, then this curve also passes through the point :

(A) $(4, 5)$
(B) $(5, 4)$
(C) $(4, 4)$
(D) $(5, 5)$

Q195·MathematicsSingle correctJEE Main 2021

Let

f

be a twice differentiable function defined on R such that

f(0)=1,\ f'(0)=2

and

f'(x)\neq 0

for all

x\in R

. If

\begin{vmatrix} f(x) & f'(x) \\ f'(x) & f''(x) \end{vmatrix}=0,

for all

x\in R

then the value of

f(1)

lies in the interval :

(A)(9, 12)
(B)(6, 9)
(C)(3, 6)
(D)(0, 3)

Q196·MathematicsSingle correctJEE Main 2021

If a curve

y=f(x)

passes through the point (1,2) and satisfies

x\frac{dy}{dx}+y=bx^{4}

, then for what value of b,

\int_{1}^{2}f(x)dx=\frac{62}{5}

?

(A)5
(B) $\frac{62}{5}$
(C) $\frac{31}{5}$
(D)10

Q197·MathematicsSingle correctJEE Main 2021

The population

P = P(t)

at time 't' of a certain species follows the differential equation

\frac{dP}{dt} = 0.5P - 450

. If

P(0) = 850

, then the time at which population becomes zero is :

(A) $\frac{1}{2}\log_e 18$
(B) $2\log_e 18$
(C) $\log_e 9$
(D) $\log_e 18$

Q198·MathematicsMultiple correctJEE Advanced 2020

Let

b

be a nonzero real number. Suppose

f : \mathbb{R} \to \mathbb{R}

is a differentiable function such that

f(0) = 1

. If the derivative

f'

of

f

satisfies the equation

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

for all

x \in \mathbb{R}

, then which of the following statements is/are TRUE?

(A)If $b > 0$ , then $f$ is an increasing function
(B)If $b < 0$ , then $f$ is a decreasing function
(C) $f(x) f(-x) = 1$ for all $x \in \mathbb{R}$
(D) $f(x) - f(-x) = 0$ for all $x \in \mathbb{R}$

Q199·MathematicsSingle correctJEE Main 2020

If

y = \left(\frac{2}{\pi}x - 1\right)\operatorname{cosec} x

is the solution of the differential equation,

\frac{dy}{dx} + p(x)y = \frac{2}{\pi}\operatorname{cosec}\ x, 0 < x < \frac{\pi}{2}

, then the function p(x) is equal to:

(A) $\cot x$
(B) $\operatorname{cosec} x$
(C) $\sec x$
(D) $\tan x$

Q200·MathematicsSingle correctJEE Main 2020

Let

y=y(x)

be the solution of the differential equation

\cos x\dfrac{dy}{dx}+2y\sin x=\sin 2x, x\in\left(0,\dfrac{\pi}{2}\right)

. If

y(\pi/3)=0

, then

y(\pi/4)

is equal to:

(A) $2+\sqrt{2}$
(B) $\dfrac{1}{\sqrt{2}}-1$
(C) $2-\sqrt{2}$
(D) $\sqrt{2}-2$

Q201·MathematicsSingle correctJEE Main 2020

If y = y (x) is the solution of the differential equation

\dfrac{5+e^{x}}{2+y}.\dfrac{dy}{dx} + e^{x} = 0

satisfying y(0) = 1, then a value of

y(\log_{e} 13)

is:

(A)0
(B)1
(C)-1
(D)2

Q202·MathematicsSingle correctJEE Main 2020

Let

y = y(x)

be the solution of the differential equation,

xy' - y = x^{2}(x \cos x + \sin x)

,

x > 0

. If

y(\pi) = \pi

, then

y''\left(\frac{\pi}{2}\right) + y\left(\frac{\pi}{2}\right)

is equal to:

(A) $1 + \frac{\pi}{2} + \frac{\pi^{2}}{4}$
(B) $2 + \frac{\pi}{2} + \frac{\pi^{2}}{4}$
(C) $2 + \frac{\pi}{2}$
(D) $1 + \frac{\pi}{2}$

Q203·MathematicsSingle correctJEE Main 2020

The solution of differential equation

\frac{dy}{dx} - \frac{y + 3x}{\log_{e}\left(y + 3x\right)} + 3 = 0

is: (where C is a constant of integration.)

(A) $y + 3x - \frac{1}{2}\left(\log_{e} x\right)^{2} = C$
(B) $x - \frac{1}{2}\left(\log_{e}\left(y + 3x\right)\right)^{2} = C$
(C) $x - 2\log_{e}\left(y + 3x\right) = C$
(D) $x - \log_{e}\left(y + 3x\right) = C$

Q204·MathematicsSingle correctJEE Main 2020

Let f : (0, ∞) → (0, ∞) 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:

(A)e
(B) $\frac{1}{e}$
(C) $\frac{1}{2e}$
(D)2e

Q205·MathematicsSingle correctJEE Main 2020

The solution curve of the differential equation,

\left(1+e^{-x}\right)\left(1+y^{2}\right)\frac{dy}{dx}=y^{2}

, which passes through the point (0, 1), is

(A) $y^{2}+1=y\left(\log_{e}\left(\frac{1+e^{x}}{2}\right)+2\right)$
(B) $y^{2}+1=y\log_{e}\left(\frac{1+e^{x}}{2}\right)$
(C) $y^{2}+1=y\left(\log_{e}\left(\frac{1+e^{-x}}{2}\right)+2\right)$
(D) $y^{2}=1+y\log_{e}\left(\frac{1+e^{x}}{2}\right)$

Q206·MathematicsSingle correctJEE Main 2020

If

x^3 dy + xy\, dx = x^2 dy + 2y\, dx; y(2) = e

and

x > 1

, then

y(4)

is equal to:

(A) $\frac{\sqrt{e}}{2}$
(B) $\frac{1}{2} + \sqrt{e}$
(C) $\frac{3}{2}\sqrt{e}$
(D) $\frac{3}{2} + \sqrt{e}$

Q207·MathematicsNumericalJEE Main 2020

If for

x \geq 0

,

y=y(x)

is the solution of the differential equation,

(x+1)dy=\left((x+1)^{2}+y-3\right)dx, y(2)=0

, then

y(3)

is equal to ______

Q208·MathematicsSingle correctJEE Main 2020

If

\dfrac{dy}{dx} = \dfrac{xy}{x^{2} + y^{2}}

;

y(1) = 1

; then a value of x satisfying

y(x) = e

is:

(A) $\sqrt{2}\,e$
(B) $\dfrac{1}{2}\sqrt{3}\,e$
(C) $\sqrt{3}\,e$
(D) $\dfrac{e}{\sqrt{2}}$

Q209·MathematicsSingle correctJEE Main 2020

The differential equation of the family of curves,

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

, is:

(A) $xy''=y'$
(B) $x\left(y'\right)^{2}=x-2yy'$
(C) $x\left(y'\right)^{2}=x+2yy'$
(D) $x\left(y'\right)^{2}=2yy'-x$

Q210·MathematicsSingle correctJEE Main 2020

Let

y=y(x)

be a solution of the differential equation,

\sqrt{1-x^{2}}\dfrac{dy}{dx}+\sqrt{1-y^{2}}=0, |x|<1

. If

y\left(\dfrac{1}{2}\right)=\dfrac{\sqrt{3}}{2}

, then

y\left(\dfrac{1}{\sqrt{2}}\right)

is equal to:

(A) $-\dfrac{\sqrt{3}}{2}$
(B) $-\dfrac{1}{\sqrt{2}}$
(C) $\dfrac{1}{\sqrt{2}}$
(D) $\dfrac{\sqrt{3}}{2}$

Q211·MathematicsSingle correctJEE Main 2020

Let

x^{k} + y^{k} = a^{k}, (a,k > 0)

and

\frac{dy}{dx} + \left(\frac{y}{x}\right)^{\frac{1}{3}} = 0

, then k is:

(A) $\frac{3}{2}$
(B) $\frac{4}{3}$
(C) $\frac{1}{3}$
(D) $\frac{2}{3}$

Q212·MathematicsSingle correctJEE Main 2020

If

y = y(x)

is the solution of the differential equation,

e^{y}\left(\frac{dy}{dx} - 1\right) = e^{x}

such that

y(0) = 0

, then

y(1)

is equal to:

(A) $1 + \log_{e} 2$
(B) $2e$
(C) $\log_{e} 2$
(D) $2 + \log_{e} 2$

Q213·MathematicsSingle correctJEE Main 2020

Let

y=y(x)

be the solution curve of the differential equation,

(y^{2}-x)\dfrac{dy}{dx}=1

, satisfying

y(0)=1

. This curve intersects the x-axis at a point whose abscissa is:

(A) $2+e$
(B) $-e$
(C)2
(D) $2-e$

Q214·MathematicsMultiple correctJEE Advanced 2019

Let

\Gamma

denote a curve

y = y(x)

which is in the first quadrant and let the point (1, 0) lie on it. Let the tangent to

\Gamma

at a point P intersect the y-axis at

Y_P

. If

PY_P

has length 1 for each point P on

\Gamma

, then which of the following option is/are correct?

(A) $y = \log_e\left(\frac{1 + \sqrt{1 - x^2}}{x}\right) - \sqrt{1 - x^2}$
(B) $xy' + \sqrt{1 - x^2} = 0$
(C) $y = -\log_e\left(\frac{1 + \sqrt{1 - x^2}}{x}\right) + \sqrt{1 - x^2}$
(D) $xy' - \sqrt{1 - x^2} = 0$

Q215·MathematicsSingle correctJEE Main 2019

Consider the differential equation,

y^{2}dx+\left(x-\frac{1}{y}\right)dy=0

. If value of y is 1 when x = 1, then the value of x for which y = 2, is :

(A) $\frac{3}{2}-\sqrt{e}$
(B) $\frac{1}{2}+\frac{1}{\sqrt{e}}$
(C) $\frac{3}{2}-\frac{1}{\sqrt{e}}$
(D) $\frac{5}{2}+\frac{1}{\sqrt{e}}$

Q216·MathematicsSingle correctJEE Main 2019

The general solution of the differential equation

(y^{2}-x^{3})dx - xy\,dy = 0

(x \neq 0)

is : (where c is a constant of integration)

(A) $y^{2}+2x^{3}+cx^{2}=0$
(B) $y^{2}-2x^{3}+cx^{2}=0$
(C) $y^{2}+2x^{2}+cx^{3}=0$
(D) $y^{2}-2x^{2}+cx^{3}=0$

Q217·MathematicsSingle correctJEE Main 2019

If y = 1f(x) is the solution of the differential equation

\frac{dy}{dx}=(\tan x - y)\sec^{2}x

, x ∈ (

-\frac{\pi}{2}, \frac{\pi}{2}

), such that y(0) = 0, then y(

-\frac{\pi}{4}

) is equal to

(A) $\frac{1}{2}-e$
(B) $\frac{1}{e}-2$
(C) $e-2$
(D) $2+\frac{1}{e}$

Q218·MathematicsSingle correctJEE Main 2019

The solution of the differential equation

x\dfrac{dy}{dx}+2y=x^{2}\ (x\neq 0)

with

y(1)=1

, is:

(A) $y=\dfrac{x^{3}}{5}+\dfrac{1}{5x^{2}}$
(B) $y=\dfrac{x^{2}}{4}+\dfrac{3}{4x^{2}}$
(C) $y=\dfrac{4}{5}x^{3}+\dfrac{1}{5x^{2}}$
(D) $y=\dfrac{3}{4}x^{2}+\dfrac{1}{4x^{2}}$

Q219·MathematicsSingle correctJEE Main 2019

If

\cos x\dfrac{dy}{dx} - y\sin x = 6x, \left(0 < x < \dfrac{\pi}{2}\right)

and

y\left(\dfrac{\pi}{3}\right) = 0

, then

y\left(\dfrac{\pi}{6}\right)

is equal to:

(A) $-\dfrac{\pi^{2}}{4\sqrt{3}}$
(B) $-\dfrac{\pi^{2}}{2}$
(C) $\dfrac{\pi^{2}}{2\sqrt{3}}$
(D) $-\dfrac{\pi^{2}}{2\sqrt{3}}$

Q220·MathematicsSingle correctJEE Main 2019

Given that the slope of the tangent to a curve

y=y(x)

at any point

(x,y)

is

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

. If the curve passes through the centre of the circle

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

, then its equation is:

(A) $x\log_{e}|y|=x-1$
(B) $x\log_{e}|y|=-2(x-1)$
(C) $x^{2}\log_{e}|y|=-2(x-1)$
(D) $x\log_{e}|y|=2(x-1)$

Q221·MathematicsSingle correctJEE Main 2019

Let

y=y(x)

be the solutions of the differential equation,

\left(x^{2}+1\right)^{2}\frac{dy}{dx}+2x\left(x^{2}+1\right)y=1

such that

y(0)=0

. If

\sqrt{a}\,y(1)=\frac{\pi}{32}

, then the value of 'a' is

(A) $\frac{1}{2}$
(B) $1$
(C) $\frac{1}{16}$
(D) $\frac{1}{4}$

Q222·MathematicsSingle correctJEE Main 2019

Let

y = y(x)

be the solution of the differential equation,

x\dfrac{dy}{dx} + y = x\log_{e} x, (x > 1)

. If

2y(2) = \log_{e} 4 - 1

, then

y(e)

is equal to:

(A) $-\dfrac{e}{2}$
(B) $-\dfrac{e^{2}}{2}$
(C) $\dfrac{e}{4}$
(D) $\dfrac{e^{2}}{4}$

Q223·MathematicsSingle correctJEE Main 2019

If a curve passes through the point (1, −2) and has slope of the tangent at any point (x, y) on it as

\dfrac{x^{2} - 2y}{x}

, then the curve also passes through the point :

(A) $(3, 0)$
(B) $(\sqrt{3}, 0)$
(C) $(-1, 2)$
(D) $(-\sqrt{2}, 1)$

Q224·MathematicsSingle correctJEE Main 2019

If y (x) is the solution of the differential equation

\frac{dy}{dx}+\left(\frac{2x+1}{x}\right)y=e^{-2x}, x>0

where

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

, then:

(A) $y\left(\log_{e}2\right)=\log_{e}4$
(B) $y\left(\log_{e}2\right)=\frac{\log_{e}2}{4}$
(C)y(x) is decreasing in $\left(\frac{1}{2},1\right)$
(D)y(x) is decreasing in (0, 1)

Q225·MathematicsSingle correctJEE Main 2019

The solution of the differential equation

\frac{dy}{dx}=(x-y)^{2}

when y(1) = 1, is:

(A) $\log_e\left|\frac{2-x}{2-y}\right|=x-y$
(B) $-\log_e\left|\frac{1-x+y}{1+x-y}\right|=2(x-1)$
(C) $-\log_e\left|\frac{1+x-y}{1-x+y}\right|=x+y-2$
(D) $\log_e\left|\frac{2-y}{2-x}\right|=2(y-1)$

Q226·MathematicsSingle correctJEE Main 2019

If

\frac{dy}{dx}+\frac{3}{\cos^{2}x}y=\frac{1}{\cos^{2}x},x\in\left(\frac{-\pi}{3},\frac{\pi}{3}\right)

and

y\left(\frac{\pi}{4}\right)=\frac{4}{3}

, then

y\left(-\frac{\pi}{4}\right)

equals:

(A) $\frac{1}{3}+e^{6}$
(B) $\frac{1}{3}$
(C) $-\frac{4}{3}$
(D) $\frac{1}{3}+e^{3}$

Q227·MathematicsSingle correctJEE Main 2019

If

y = y(x)

is solution of the differential equation

x\dfrac{dy}{dx} + 2y = x^2

satisfying

y(1) = 1

, then

y\left(\dfrac{1}{2}\right)

is equal to

(A) $\dfrac{7}{64}$
(B) $\dfrac{1}{4}$
(C) $\dfrac{49}{16}$
(D) $\dfrac{13}{16}$

Q228·MathematicsSingle correctJEE Main 2019

Let

f:[0,1]\rightarrow R

be such that

f(xy)=f(x)f(y)

for all

x,y\in[0,1]

, and

f(0)\neq 0

. If

y=y(x)

satisfies the differential equation,

\frac{dy}{dx}=f(x)

with

y(0)=1

, then

y\left(\frac{1}{4}\right)+y\left(\frac{3}{4}\right)

is equal to

(A)4
(B)3
(C)5
(D)2

Q229·MathematicsNumericalJEE Advanced 2018

Let

f : R \to R

be a differentiable function with

f(0) = 0

. If

y = f(x)

satisfies the differential equation

\frac{dy}{dx} = (2+5y)(5y-2)

, then the value of

\lim_{x \to -\infty} f(x)

is ______ .

Q230·MathematicsMultiple correctJEE Advanced 2018

Let

f : [0, \infty) \to R

be a continuous function such that

f(x) = 1 - 2x + \int_{0}^{x} e^{x-t} f(t) dt

for all

x \in [0, \infty)

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

(A)The curve $y = f(x)$ passes through the point $(1, 2)$
(B)The curve $y = f(x)$ passes through the point $(2, -1)$
(C)The area of the region $\{(x, y) \in [0, 1] \times R : f(x) \le y \le \sqrt{1-x^{2}}\}$ is $\frac{\pi-2}{4}$
(D)The area of the region $\{(x, y) \in [0, 1] \times R : f(x) \le y \le \sqrt{1-x^{2}}\}$ is $\frac{\pi-1}{4}$

Q231·MathematicsNumericalJEE Advanced 2018

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))

is ______ .

Q232·MathematicsMultiple correctJEE Advanced 2018

Let

f : (0, \pi) \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, \pi)

If

f\left(\frac{\pi}{6}\right) = -\frac{\pi}{12}

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

(A) $f\left(\frac{\pi}{4}\right) = \frac{\pi}{4\sqrt{2}}$
(B) $f(x) < \frac{x^{4}}{6} - x^{2}$ for all $x \in (0, \pi)$
(C)There exists $\alpha \in (0, \pi)$ such that $f'(\alpha) = 0$
(D) $f''\left(\frac{\pi}{2}\right) + f\left(\frac{\pi}{2}\right) = 0$

Q233·MathematicsMultiple correctJEE Advanced 2018

Let

f : R \to R

and

g : R \to R

be two non-constant differentiable functions. If

f'(x) = \left(e^{(f(x)-g(x))}\right) g'(x)

for all

x \in R

, and

f(1) = g(2) = 1

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

(A) $f(2) < 1 - \log_{e} 2$
(B) $f(2) > 1 - \log_{e} 2$
(C) $g(1) > 1 - \log_{e} 2$
(D) $g(1) < 1 - \log_{e} 2$

Q234·MathematicsSingle correctJEE Advanced 2017

If

y = y(x)

satisfies the differential equation

8\sqrt{x}\left(\sqrt{9+\sqrt{x}}\right)dy = \left(\sqrt{4+\sqrt{9+\sqrt{x}}}\right)^{-1}dx

,

\quad x > 0

and

y(0) = \sqrt{7}

, then

y(256) =

(A) $3$
(B) $9$
(C) $16$
(D) $80$

Q235·MathematicsMultiple correctJEE Advanced 2016

A solution curve of the differential equation

\left(x^{2} + xy + 4x + 2y + 4\right)\frac{dy}{dx} - y^{2} = 0

,

x > 0

, passes through the point (1, 3). Then the solution curve

(A)intersects $y = x + 2$ exactly at one point
(B)intersects $y = x + 2$ exactly at two points
(C)intersects $y = (x + 2)^{2}$
(D)does NOT intersect $y = (x + 3)^{2}$

Q236·MathematicsMultiple correctJEE Advanced 2016

Let

f : (0, \infty) \to \mathbb{R}

be a differentiable function such that

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

for all

x \in (0, \infty)

and

f(1) \neq 1

. Then

(A) $\lim_{x \to 0+} f'\left(\frac{1}{x}\right) = 1$
(B) $\lim_{x \to 0+} x f\left(\frac{1}{x}\right) = 2$
(C) $\lim_{x \to 0+} x^{2} f'(x) = 0$
(D) $|f(x)| \leq 2$ for all $x \in (0, 2)$

Q237·MathematicsMultiple correctJEE Advanced 2015

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

Py'' + Qy' + 1 = 0

, where

P

,

Q

are functions of

x

,

y

and

y'

(here

y' = \dfrac{dy}{dx}

,

y'' = \dfrac{d^{2}y}{dx^{2}}

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

(A) $P = y + x$
(B) $P = y - x$
(C) $P + Q = 1 - x + y + y' + (y')^{2}$
(D) $P - Q = x + y - y' - (y')^{2}$

Q238·MathematicsMultiple correctJEE Advanced 2015

Let

y(x)

be a solution of the differential equation

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

. If

y(0) = 2

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

(A) $y(-4) = 0$
(B) $y(-2) = 0$
(C) $y(x)$ has a critical point in the interval $(-1, 0)$
(D) $y(x)$ has no critical point in the interval $(-1, 0)$

Q239·MathematicsSingle correctJEE Advanced 2014

The function

y = f(x)

is the solution of the differential equation

\frac{dy}{dx} + \frac{xy}{x^{2}-1} = \frac{x^{4}+2x}{\sqrt{1-x^{2}}}

in

(-1, 1)

satisfying

f(0) = 0

. Then

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

is

(A) $\frac{\pi}{3} - \frac{\sqrt{3}}{2}$
(B) $\frac{\pi}{3} - \frac{\sqrt{3}}{4}$
(C) $\frac{\pi}{6} - \frac{\sqrt{3}}{4}$
(D) $\frac{\pi}{6} - \frac{\sqrt{3}}{2}$

Q240·MathematicsSingle correctJEE Advanced 2013

A curve passes through the point

(1,\frac{\pi}{6})

. Let the slope of the curve at each point

(x,y)

be

\frac{y}{x}+\sec\left(\frac{y}{x}\right)

,

x>0

. Then the equation of the curve is

(A) $\sin\left(\frac{y}{x}\right)=\log x+\frac{1}{2}$
(B) $\operatorname{cosec}\left(\frac{y}{x}\right)=\log x+2$
(C) $\sec\left(\frac{2y}{x}\right)=\log x+2$
(D) $\cos\left(\frac{2y}{x}\right)=\log x+\frac{1}{2}$

Differential Equations — JEE Previous Year Questions

All 240 Differential Equations questions

Differential Equations — frequently asked

How many questions from Differential Equations appear in JEE?

Is Differential Equations an important chapter for JEE?

Where do these Differential Equations questions come from?

Other Mathematics chapters

Practise Differential Equations until it stops costing you marks.