Core Syllabus Module

Differential Equations

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Chapter Elite-20 Challenge Zone

Target Advanced

JEE Advanced 2018

LEVELJEE Advanced

Let $f : (0, π) \to R$ be a twice differentiable function such that $lim_{t \to x} \frac{f ( x ) s i n t - f ( t ) s i n x}{t - x} = sin^{2} x$ for all $x \in (0, π)$ . If $f (\frac{π}{6}) = - \frac{π}{12}$ , then which of the following statement(s) is (are) TRUE ?

* Multiple Correct Options

(A)

\int_{0}^{π /2} f (x) d x = \frac{π ^{2}}{4 2}

(B)

f (x) < - \frac{x ^{2}}{6}

for all

x \in (0, π)

(C)

There exists

α \in (0, π)

such that

f^{'} (α) = 0

(D)

f^{''} (\frac{π}{2}) + f (\frac{π}{2}) = 0

JEE Main 2026 (21 January Shift 1)

LEVELJEE Advanced

Let $f : R \to R$ be a twice differentiable function such that the quadratic equation $f (x) m^{2} - 2 f^{'} (x) m + f^{''} (x) = 0$ in $m$ , has two equal roots for every $x \in R$ . If $f (0) = 1, f^{'} (0) = 2$ and $(α, β)$ is the largest interval in which the function $f (lo g_{e} x - x)$ is increasing, then $α + β$ is equal to .........

JEE(ADVANCED)-201

LEVELJEE Advanced

Let $f : [0, \infty) \to R$ be a continuous function such that $f (x) = 1 - 2 x + \int_{0}^{x} e^{x - t} f (t) d t$ for all $x \in [0, \infty)$ . Then, which of the following statement(s) is (are) TRUE ?

* Multiple Correct Options

(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) \leq y \leq 1 - x^{2}}

\frac{π - 2}{4}

(D)

The area of the region

{(x, y) \in [0, 1] \times R : f (x) \leq y \leq 1 - x^{2}}

\frac{π - 1}{4}

JEE Advanced 2018

LEVELJEE Advanced

Let $f : R \to R$ be a differentiable function with $f (0) = 0$ . If $y = f (x)$ satisfies the differential equation $\frac{d y}{d x} = (2 + 5 y) (y - 2)$ , then the value of $lim_{x \to \infty} f (x)$ is ________.

JEE Advanced 2006

LEVELJEE Advanced

Match the following :

List-I

(P)

\int_{0}^{π /2} (sin x)^{c o s x} (cos x cot x - lo g (sin x)^{s i n x}) d x

(Q)

Area bounded by

- 4 y^{2} = x

and

x - 1 = - 5 y^{2}

(R)

Cosine of the angle of intersection of curves

y = 3^{x - 1} lo g x

and

y = x^{x} - 1

(S)

Let

\frac{d y}{d x} = \frac{6}{x + y}

where

y (0) = 0

then value of

y

when

x + y = 6

List-II

(1)

(2)

(3)

6 ln 2

(4)

4/3

JEE Advanced 2001

LEVELJEE Advanced

A hemispherical tank of radius 2 metres is initially full of water and has an outlet of $12 cm^{2}$ cross-sectional area at the bottom. The outlet is opened at some instant. The flow through the outlet is according to the law $v (t) = 0.6 2 g h (t)$ , where $v (t)$ and $h (t)$ are respectively the velocity of the flow through the outlet and the height of water level above the outlet at time t, and g is the acceleration due to gravity. Find the time it takes to empty the tank.

JEE Advanced 2009

LEVELJEE Advanced

Match the statements/expressions given in Column-I with the values given in Column-II.

List-I

(P)

The number of solutions of the equation

x e^{s i n x} - cos x = 0

in the interval

(0, \frac{π}{2})

(Q)

Value(s) of

k

for which the planes

k x + 4 y + z = 0

4 x + k y + 2 z = 0

and

2 x + 2 y + z = 0

intersect in a straight line

(R)

Value(s) of

k

for which

∣ x - 1∣ + ∣ x - 2∣ + ∣ x + 1∣ + ∣ x + 2∣ = 4 k

has integer solution(s)

(S)

y^{'} = y + 1

and

y (0) = 1

, then value(s) of

y (ln 2)

List-II

JEE Main 2025 (January)

LEVELJEE Advanced

Let $y = f (x)$ be the solution of the differential equation $\frac{d y}{d x} + \frac{x y}{x ^{2} - 1} = \frac{x ^{6} + 4 x}{1 - x ^{2}},$ $- 1 < x < 1$ such that $f (0) = 0$ . If $6 \int_{- 1/2}^{1/2} f (x) d x = 2 π - α$ then $α^{2}$ is equal to

JEE Main 2021 (20 July Shift 1)

LEVELJEE Advanced

Let $y = y (x)$ be the solution of the differential equation $x tan (\frac{y}{x}) d y = (y tan (\frac{y}{x}) - x) d x$ , $- 1 \leq x \leq 1, y (\frac{1}{2}) = \frac{π}{6}$ . Then the area of the region bounded by the curves $x = 0, x = \frac{1}{2}$ and $y = y (x)$ in the upper half plane is:

(A)

\frac{1}{8} (π - 1)

(B)

\frac{1}{12} (π - 3)

(C)

\frac{1}{4} (π - 2)

(D)

\frac{1}{6} (π - 1)

JEE Main 2022 (26 July Shift 1)

LEVELJEE Advanced

Let a curve $y = y (x)$ 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 $(\frac{y}{x})^{3}$ . If this curve also passes through the point $(α, 610)$ in the first quadrant, then $α$ is equal to

JEE Main 2021 (February)

LEVELJEE Advanced

If a curve $y = f (x)$ passes through the point (1,2) and satisfies $x \frac{d y}{d x} + y = b x^{4}$ , then for what value of b, $\int_{1}^{2} f (x) d x = \frac{62}{5}$ ?

(A)

(B)

\frac{62}{5}

(C)

\frac{31}{5}

(D)

JEE Advanced 2019

LEVELJEE Advanced

Let $Γ$ 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 $Γ$ at a point $P$ intersect the y-axis at $Y_{P}$ . If $P Y_{P}$ has length 1 for each point $P$ on $Γ$ , then which of the following options is/are correct?

* Multiple Correct Options

(A)

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

(B)

x y^{'} - 1 - x^{2} = 0

(C)

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

(D)

x y^{'} + 1 - x^{2} = 0

JEE Main 2021 (18 March Shift 2)

LEVELJEE Advanced

Let $y = y (x)$ be the solution of the differential equation $x d y - y d x = x^{2} - y^{2} d x, x \geq 1$ , with $y (1) = 0$ . If the area bounded by the line $x = 1, x = e^{π}, y = 0$ and $y = y (x)$ is $α e^{2 π} + β$ , then the value of $10 (α + β)$ is equal to ____.

JEE Main 2021 (17 March Shift 2)

LEVELJEE Advanced

Let $y = y (x)$ be the solution of the differential equation $cos x (3 sin x + cos x + 3) d y = (1 + y sin x (3 sin x + cos x + 3)) d x$ , $0 \leq x \leq \frac{π}{2}, y (0) = 0$ . Then $y (\frac{π}{3})$ is equal to:

(A)

2 lo g_{e} (\frac{2 3 + 9}{6})

(B)

2 lo g_{e} (\frac{2 3 + 10}{11})

(C)

2 lo g_{e} (\frac{3 + 7}{2})

(D)

2 lo g_{e} (\frac{3 3 - 8}{4})

JEE Main 2026 (22 January Shift 2)

LEVELJEE Advanced

If $y = y (x)$ satisfies the differential equation $16 x + 9 x (4 + 9 + x) cos y d y = (1 + 2 sin y) d x, x > 0$ and $y (256) = \frac{π}{2}, y (49) = α$ , then $2 sin α$ is equal to :

(A)

2 (2 - 1)

(B)

3 (2 - 1)

(C)

22 - 1

(D)

2 - 1

JEE Main 2021 (18 March Shift 2)

LEVELJEE Advanced

Let $y = y (x)$ be the solution of the differential equation $\frac{d y}{d x} = (y + 1) ((y + 1) e^{x^{2} /2} - x), 0 < x < 2.1$ , with $y (2) = 0$ . Then the value of $\frac{d y}{d x}$ at $x = 1$ is equal to:

(A)

\frac{- e ^{3/2}}{( e ^{2} + 1 ) ^{2}}

(B)

- \frac{2 e ^{2}}{( 1 + e ^{2} ) ^{2}}

(C)

\frac{e ^{5/2}}{( 1 + e ^{2} ) ^{2}}

(D)

\frac{5 e ^{1/2}}{( e ^{2} + 1 ) ^{2}}

JEE Main 2022 (28 June Shift 2)

LEVELJEE Advanced

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 $(π /4, 0)$ , then the value of $\int_{0}^{π /2} y d x$ is equal to

(A)

(2 - 2) + \frac{π}{2}

(B)

2 - \frac{π}{2}

(C)

(2 + 2) + \frac{π}{2}

(D)

2 + \frac{π}{2}

JEE Main 2026 (24 January Shift 1)

LEVELJEE Advanced

Let a differentiable function $f$ satisfy the equation $\int_{0}^{36} f (\frac{t x}{36}) d t = 4 α f (x)$ . If $y = f (x)$ is a standard parabola passing through the points $(2, 1)$ and $(- 4, β)$ , then $β^{α}$ is equal to .........

JEE Main 2026 (28 January Shift 2)

LEVELJEE Advanced

Let $f$ be a differentiable function satisfying $f (x) = 1 - 2 x + \int_{0}^{x} e^{(x - t)} f (t) d t, x \in R$ and let $g (x) = \int_{0}^{x} (f (t) + 2)^{15} (t - 4)^{6} (t + 12)^{17} d t, x \in R$ . If $p$ and $q$ are respectively the points of local minima and local maxima of $g$ , then the value of $∣ p + q ∣$ is equal to .........

JEE Main 2021 (22 July Shift 1)

LEVELJEE Advanced

Let $y = y (x)$ be the solution of the differential equation $((x + 2) e^{\frac{y + 1}{x + 2}} + (y + 1)) d x = (x + 2) d y, y (1) = 1$ . If the domain of $y = y (x)$ is an open interval $(α, β)$ , then $∣ α + β ∣$ is equal to ____.

Micro-Topic Strategy Matrix

Order and Degree of a Differential Equation

Formation of Differential Equations

Variable Separable Method

Homogeneous Differential Equations

Linear Differential Equations

Orthogonal Trajectories

Chapter Elite-20 Challenge Zone

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let f:[0,∞)→R be a continuous function such that f(x)=1−2x+∫0x​ex−tf(t)dt for all x∈[0,∞). Then, which of the following statement(s) is (are) TRUE ?

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let f:R→R be a differentiable function with f(0)=0. If y=f(x) satisfies the differential equation dxdy​=(2+5y)(y−2), then the value of limx→∞​f(x) is ________.

Match the following :

List-I

List-II

Match the statements/expressions given in Column-I with the values given in Column-II.

List-I

List-II

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If a curve y=f(x) passes through the point (1,2) and satisfies xdxdy​+y=bx4, then for what value of b, ∫12​f(x)dx=562​?

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let y=y(x) be the solution of the differential equation cosx(3sinx+cosx+3)dy=(1+ysinx(3sinx+cosx+3))dx, 0≤x≤2π​,y(0)=0. Then y(3π​) is equal to:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If y=y(x) satisfies the differential equation 16x+9x​​(4+9+x​​)cosydy=(1+2siny)dx,x>0 and y(256)=2π​,y(49)=α, then 2sinα is equal to :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let y=y(x) be the solution of the differential equation dxdy​=(y+1)((y+1)ex2/2−x),0<x<2.1, with y(2)=0. Then the value of dxdy​ at x=1 is equal to:

Let $f : [0, \infty) \to R$ be a continuous function such that $f (x) = 1 - 2 x + \int_{0}^{x} e^{x - t} f (t) d t$ for all $x \in [0, \infty)$ . Then, which of the following statement(s) is (are) TRUE ?

Let $f : R \to R$ be a differentiable function with $f (0) = 0$ . If $y = f (x)$ satisfies the differential equation $\frac{d y}{d x} = (2 + 5 y) (y - 2)$ , then the value of $lim_{x \to \infty} f (x)$ is ________.

If a curve $y = f (x)$ passes through the point (1,2) and satisfies $x \frac{d y}{d x} + y = b x^{4}$ , then for what value of b, $\int_{1}^{2} f (x) d x = \frac{62}{5}$ ?

Let $y = y (x)$ be the solution of the differential equation $cos x (3 sin x + cos x + 3) d y = (1 + y sin x (3 sin x + cos x + 3)) d x$ , $0 \leq x \leq \frac{π}{2}, y (0) = 0$ . Then $y (\frac{π}{3})$ is equal to:

If $y = y (x)$ satisfies the differential equation $16 x + 9 x (4 + 9 + x) cos y d y = (1 + 2 sin y) d x, x > 0$ and $y (256) = \frac{π}{2}, y (49) = α$ , then $2 sin α$ is equal to :

Let $y = y (x)$ be the solution of the differential equation $\frac{d y}{d x} = (y + 1) ((y + 1) e^{x^{2} /2} - x), 0 < x < 2.1$ , with $y (2) = 0$ . Then the value of $\frac{d y}{d x}$ at $x = 1$ is equal to: