Differential Equations — JEE Maths Practice Questions & Solutions

hardPYQ · JEE Main 2026

Let

y=y(x)

be the solution of the differential equation

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

,

x\ge1

. If

y(1)=1

, then the greatest integer less than

y\left(\sqrt5\right)

is

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hardPYQ · JEE Main 2026

Let

y=y(x)

be the solution of

\dfrac{dy}{dx}+\left(\dfrac{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)=\dfrac32

. If

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

, then

\alpha

is equal to

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mediumPYQ · JEE Main 2026

Let

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

be defined as

f(x)=\displaystyle\int_1^x f(t)\,dt+(1-x)(\ln x-1)+e

. Then the value of

f(f(1))

is:

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mediumPYQ · JEE Main 2026

Let

y=y(x)

satisfy

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

on

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

, with

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

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mediumPYQ · JEE Main 2026

Let

x=x(y)

be the solution of the differential equation

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

,

y>1

,

x(e)=e

. Then

x(e^2)

is equal to

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mediumPYQ · JEE Main 2026

Let

y=y(x)

be the solution of

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

,

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

, and let

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

. Then the number of integral values of

p

for which

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

represents a circle of radius

r\le6

is

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mediumPYQ · JEE Main 2026

Let

y=y(x)

be the solution of the differential equation

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

,

y(0)=\dfrac12

. Then

(2y(1)-1)

is equal to

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mediumPYQ · JEE Main 2026

Let

y=y(x)

be the solution curve of the differential equation

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

,

y(0)=0

. If the curve

y=y(x)

passes through the point

\left(\alpha,-\dfrac12\right)

, then a value of

\alpha

is

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mediumPYQ · JEE Main 2026

If the curve

y=f(x)

passes through the point

(1,e)

and satisfies the differential equation

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

,

x>0

, then

f(e)

is equal to

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hard

Let the curve

y = f(x)

pass through the origin and satisfy

\dfrac{dy}{dx} + \displaystyle\int_0^5 y\,dx = 27

. If

a

and

b

are chosen randomly from

S = \{1, 2, 3, 4\}

with replacement, the probability that the curve passes through

(a, b)

is:

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hard

Let

y'(x) + \dfrac{g'(x)}{g(x)}\,y(x) = \dfrac{g'(x)}{1 + g^2(x)}

, where

g(x)

is a given non-constant differentiable function on

R

. If

g(1) = y(1) = 1

and

g(e) = \sqrt{2e - 1}

, then

y(e)

equals:

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hard

Suppose a solution of

(xy^3 + x^2y^7)\dfrac{dy}{dx} = 1

satisfies

y\left(\dfrac{1}{4}\right) = 1

. Then the value of

\dfrac{dy}{dx}

when

y = -1

is:

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medium

The degree of the differential equation satisfying the relation

\sqrt{1+x^2} + \sqrt{1+y^2} = \lambda\big(x\sqrt{1+y^2} - y\sqrt{1+x^2}\big)

is:

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medium

If

p

and

q

are the order and degree of the differential equation

y^2\left(\dfrac{d^2y}{dx^2}\right)^2 + 3x\left(\dfrac{dy}{dx}\right)^{1/3} + x^2y^2 = \sin x

, then:

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medium

A curve, whose concavity is directly proportional to the logarithm of its

x

-coordinate at any point of the curve, is given by:

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medium

The solution of the differential equation

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

is:

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medium

The solution of the differential equation

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

, if

y(1) = 0

, is:

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medium

If a curve

y = f(x)

satisfies

y''x^2 + x(y')^2 = 2xy'

, with

f(0) = 0

and

f'(1) = 1

, then

f(x)

is:

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medium

A continuous function

f: R \to R

satisfies

f(x) = (1+x^2)\left[1 + \displaystyle\int_0^x \dfrac{f^2(t)}{1+t^2}\,dt\right]

. Then the value of

f(-2)

is:

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medium

The solution of the differential equation

x\,dx + y\,dy + \dfrac{x\,dy - y\,dx}{x^2 + y^2} = 0

is:

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medium

The solution of

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

is:

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medium

The solution of the differential equation

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

is:

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medium

The primitive of the differential equation

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

is:

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medium

A function

y = f(x)

satisfies

f'(x)\sin x + f(x)\cos x = 1

, with

f(x)

bounded as

x \to 0

. If

I = \displaystyle\int_0^{\pi/2} f(x)\,dx

, then:

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medium

A tangent drawn at any point

P

on a curve meets the

x

-axis at

Q

such that the circumcentre of

\triangle POQ

has abscissa half that of its ordinate. The differential equation of such a curve is:

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medium

The solution of

x\sec\left(\dfrac{y}{x}\right)(y\,dx + x\,dy) = y\,\mathrm{cosec}\left(\dfrac{y}{x}\right)(x\,dy - y\,dx)

is:

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medium

If the independent variable

x

is changed to

y

, then the differential equation

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

is transformed to

x\dfrac{d^2x}{dy^2} + \left(\dfrac{dx}{dy}\right)^2 = k

, where

k

is a number. Then

k

equals:

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medium

The solution of

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

is:

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medium

The solution of the differential equation

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

is:

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medium

If the solution of

\dfrac{dy}{dx} = y + \displaystyle\int_0^2 y\,dx

is

y(x)

with

y(0) = 1

, then

\big[\,|y(2)|\,\big]

equals (where

[\,\cdot\,]

is the greatest integer function):

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medium

The solution of the differential equation

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

is given by:

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medium

Let

y = y(x)

and

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

. If

y(1) = 1

, then the value of

\big[\,y\left(\tfrac{1}{e}\right)\big]

is (where

[\,\cdot\,]

is the greatest integer function):

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medium

If the substitution

x = \tan^{-1}(t)

transforms the differential equation

\dfrac{d^2y}{dx^2} + xy\dfrac{dy}{dx} + \sec^2 x = 0

into

(1 + t^2)\dfrac{d^2y}{dt^2} + \big(2t + y\tan^{-1}(t)\big)\dfrac{dy}{dt} = k

, then the value of

k

is:

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medium

A function

y = f(x)

satisfies

x = \dfrac{dy}{dx} - \dfrac{1}{2}\left(\dfrac{dy}{dx}\right)^2 + \dfrac{1}{3}\left(\dfrac{dy}{dx}\right)^3 - \cdots\infty

, with

y(0) = 1

and

x \le \ln 2

. Then

f'\left(\ln\dfrac{1}{2}\right) + f\left(\ln\dfrac{1}{2}\right)

equals:

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medium

If

y_1(x)

is a solution of

\dfrac{dy}{dx} - f(x)\,y = 0

, then a solution of

\dfrac{dy}{dx} + f(x)\,y = r(x)

is:

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medium

The equation of the curve for which the square of the ordinate is twice the rectangle contained by the abscissa and the intercept of the normal on the

x

-axis, and passing through

(2, 1)

, is:

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medium

The largest value of

c

such that there exists a differentiable function

h(x)

for

-c < x < c

that is a solution of

y' = 1 + y^2

with

h(0) = 0

is:

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medium

The solution of

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

is (where

c

is an arbitrary constant)

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medium

The solution of

\dfrac{dy}{dx} = \dfrac{x - 2y + 3}{2x - y + 5}

is

f(x, y, c) = 0

. For suitable values of

c

,

f(x, y, c) = 0

represents a pair of lines whose point of intersection is:

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medium

The solution of

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

is

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medium

A normal at any point

(x, y)

to the curve

y = f(x)

cuts a triangle of unit area with the axes. The differential equation of the curve is:

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easy

The order of the differential equation corresponding to

y = c_1\cos 2x + c_2\cos^2 x + c_3\sin^2 x + c_4

is:

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easy

The differential equation of all parabolas with axis parallel to the

y

-axis is:

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easy

Let

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

for all

x \in R - \{0\}

with

y(1) = e - 1

. If

y(2) = k\,y(1)\big(y(1)+2\big)

, then the value of

k

is:

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easy

The equation

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

, with

y(1) = 1

, is the differential equation of:

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easy

The differential equation

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

has the particular solution

y(0) = 1

. The value of

x

when

y = 0

is:

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easy

If

f''(x) = -f(x)

, then

\dfrac{[f(2)]^2 + [f'(2)]^2}{[f(4)]^2 + [f'(4)]^2}

equals:

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easy

A tangent to a curve intersects the

y

-axis at a point

P

. A line perpendicular to this tangent through

P

passes through another point

(1, 0)

. The differential equation of the curve is:

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easy

If

y = y(x)

and

\left(\dfrac{2 + \sin x}{y + 1}\right)\dfrac{dy}{dx} = -\cos x

with

y(0) = 0

, then

y\left(\dfrac{5\pi}{6}\right)

equals:

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easy

Let

y = y(x)

satisfy

y(x) + \displaystyle\int_1^x y(t)\,dt = x^2

. The value of

y(e)

is:

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Differential Equations — JEE Maths Practice Questions & Solutions

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