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JEE • MATHEMATICS • DIFFERENTIAL EQUATIONS

Differential equations practice · Hard

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Practice format and chapter quiz

These sets use single-correct-answer MCQs. They do not cover every JEE Main or JEE Advanced question format and are not full-length mock papers.

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1 Hard

1 of 29

Application +4 | -1

Let

y=y_{1}(x)

and

y=y_{2}(x)

be the solution curves of the differential equation

\frac{d y}{d x}=y+7

with initial conditions

y_{1}(0)=0

and

y_{2}(0)=1

respectively. Then the curves

y=y_{1}(x)

and

y=y_{2}(x)

intersect at

View answer and explanation

Correct option: A

No point

Option A (No point) is correct because solving the differential equation and applying the initial conditions leads to two lines that never intersect.

2 Hard

2 of 29

Application +4 | -1

y'' - 3y' + 2y = 0

where

y\left( 0 \right) = 1,y'\left( 0 \right) = 0

, then the value of

y

x = \log 2

View answer and explanation

Correct option: D

Option D (0) is correct because substituting

x = \log 2

into the solution of the differential equation, obtained using the given initial conditions, yields

y = 0

3 Hard

3 of 29

Application +4 | -1

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

, then

\sin \frac{y}{x}

is equal to

View answer and explanation

Correct option: B

cx

Option B (

cx

) is correct because it is the result of solving the homogeneous differential equation using the substitution method.

4 Hard

4 of 29

Application +4 | -1

The solution of the differential equation

\left(1+y^2\right)+\left(x-e^{\operatorname{tan} x^{-1} y}\right) \frac{d y}{d x}=0

is ...... .

View answer and explanation

Correct option: A

2 x e^{\tan ^{-1} y}=e^{2 \tan ^{-1} y}+c

Option A (

2 x e^{\tan ^{-1} y}=e^{2 \tan ^{-1} y}+c

) is the correct solution, found by rearranging the equation and using an integrating factor.

5 Hard

5 of 29

Application +4 | -1

The rate of decay of a radioactive body at any time is proportional to its mass present at that time. If the mass of body is

100 \mathrm{gms}

after one hour and

80 \mathrm{gms}

after two hours, then the initial mass of radioactive body is ........ .

View answer and explanation

Correct option: C

125

Option C (125) is correct because it is the initial mass calculated using the exponential decay model and the given data points.

6 Hard

6 of 29

Application +4 | -1

Let

u(x)

and

v(x)

are differentiable functions such that

\frac{u(x)}{v(x)}=7

. If

\frac{u^{\prime}(x)}{v^{\prime}(x)}=p

and

\left(\frac{u(x)}{v(x)}\right)=\mathrm{q}

, then

\frac{p+q}{p-q}

has the value equal to :

View answer and explanation

Correct option: A

Option A (1) is correct because after differentiating and simplifying, the value of (p+q)/(p-q) is found to be 1.

7 Hard

7 of 29

Application +4 | -1

\left(2 x y-y^2-y\right) d x=\left(2 x y+x-x^2\right) d y

and

y(1)=1

, then the value of

12|y(-1)|

View answer and explanation

Correct option: A

The correct answer is 12 because solving the differential equation with the given initial condition leads to y(-1) = -1, and 12*|-1| = 12.

8 Hard

8 of 29

Application +4 | -1

The differential equation of all straight lines touching the circle

{x^2} + {y^2} = {a^2}

View answer and explanation

Correct option: B

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

Option B (

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

) is correct because it is the differential equation representing all straight lines tangent to the circle

{x^2} + {y^2} = {a^2}

9 Hard

9 of 29

Application +4 | -1

The general solution of differential equation

\frac{dy}{dx}+\frac{y\ell ny}{x}=\frac{y{{(\ell ny)}^{2}}}{{{x}^{2}}}

(where

\mathrm{C}

is an arbitrary constant)

View answer and explanation

Correct option: C

\frac{1}{\ell n y}=\frac{1}{2 x}+C x

Option C (

\frac{{1}}{{\ell \text{{ ny }}}} = \frac{{1}}{{2 x}} + C x

) is correct because it is the general solution obtained after solving the given differential equation using a substitution and integrating factor.

10 Hard

10 of 29

Analyzing +4 | -1

The normal at any point

P(x, y)

of a curve meets

x

- and

y

-axes in

A

and

B

, respectively. If

O

is the origin and

O A

and

O B

are the algebraic intercepts of the normal at

P

such that

\frac{1}{O A}+\frac{1}{O B}=1

then the equation of the curve is [given that

(5,4)

is a point on the curve]

View answer and explanation

Correct option: D

(x-1)^2+(y-1)^2=25

Option D (

(x-1)^2+(y-1)^2=25

) is correct because it represents the equation of the curve obtained by solving the given differential equation and using the point (5,4).

11 Hard

11 of 29

Application +4 | -1

In an experiment of culture of bacteria in a laboratory, the rate of increase of bacteria is proportional to the number of bacteria present at that line. If at the end of one hour the number of bacteria gets doubled, then the number of bacteria at the end of 3 hours is

View answer and explanation

Correct option: B

Option B (8) is correct because the number of bacteria triples every hour, so after 3 hours, it will be 8 times the initial amount.

12 Hard

12 of 29

Analyzing +4 | -1

The differential equation satisfied by the family of curves

y = ax\cos \left( {\frac{1}{x} + b} \right)

where

a,b

are parameters is

View answer and explanation

Correct option: B

{x^4}{y_2} + y = 0

Option B (

{x^4}{y_2} + y = 0

) is correct because it is the differential equation obtained by differentiating the given family of curves twice and eliminating the parameters a and b.

13 Hard

13 of 29

Application +4 | -1

A rumour spreeds through a population of 5000 people at a rate proportional to the product of the number of people who have heard it and the number who have not. Suppose that 100 people initiate the rumour and that a total of 500 people know the rumour after 2 days. How long will it take for half the people to hear the rumour.

(\log 9/\log 49=129 / 229)

View answer and explanation

Correct option: A

Option A (5) is correct because after solving the differential equation and using the given conditions, it takes approximately 5 days for half the people to hear the rumor.

14 Hard

14 of 29

Analyzing +4 | -1

y=(1+x)\left(x^{2}+1\right)\left(x^{4}+1\right)\left(x^{8}+1\right)\left(x^{16}+1\right)

, then find the value of

y^{\prime \prime}-y^{\prime}

x=-1

View answer and explanation

Correct option: C

-496

Option C (-496) is correct because after simplifying the expression for y and differentiating twice, substituting x = -1 yields y'' - y' = -496.

15 Hard

15 of 29

Application +4 | -1

The solution of the differential equation

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

View answer and explanation

Correct option: A

a y^2=e^{x^2 / y^2}

Option A (

a y^2=e^{x^2 / y^2}

) is the correct solution to the given differential equation, which is solved by substituting

y=vx

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16 Hard

16 of 29

Application +4 | -1

y = a{x^{n + 1}}

, then

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

is equal to

View answer and explanation

Correct option: B

n\left( {n + 1} \right)y

Option B (

n\left( {n + 1} \right)y

) is correct because it is the result of calculating the second derivative and multiplying by

x^2

17 Hard

17 of 29

Application +4 | -1

Let a differentiable function

f

satisfy

f(x)+\int\limits_{{3}}^{x} \frac{f(t)}{t} d t=\sqrt{x+1}, x \geq 3

. Then

12 f(8)

is equal to

View answer and explanation

Correct option: C

The correct answer is C (17) because it is the value of 12f(8) obtained by solving the given differential equation.

18 Hard

18 of 29

Application +4 | -1

The general solution of the differential equation

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

is given by

View answer and explanation

Correct option: C

y = \left( {{c_1} + {c_2}x} \right){e^{ - x}} + \frac{{{e^{3x}}}}{8}

Option C (

y = \left( {{c_1} + {c_2}x} \right){e^{ - x}} + \frac{{{e^{3x}}}}{8}

) is correct because it represents the general solution of the given second-order linear homogeneous differential equation with constant coefficients.

19 Hard

19 of 29

Application +4 | -1

The solution of the differential equation

2 y e^{\frac{x}{y}} d x-\left(y+2 x e^{\frac{y}{y}}\right) d y=0

View answer and explanation

Correct option: A

2 c e^{\frac{x}{y}}=\log y

Option A (

2 c e^{\frac{x}{y}}=\log y

) is the correct solution, found by recognizing the equation as homogeneous and using the appropriate substitution.

20 Hard

20 of 29

Application +4 | -1

Solution of differential equation

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

View answer and explanation

Correct option: A

\log _e\left(x^2+y^2\right)+2 \tan ^{-1} \frac{y}{x}+c=0

Option A (

\log _e(x^2+y^2)+2 \tan ^{-1} \frac{y}{x}+c=0

) is the correct solution obtained by solving the given homogeneous differential equation using the substitution

y=vx

21 Hard

21 of 29

Application +4 | -1

If the curve

y(x)

satisfying

x d x=\left(\frac{x^2}{y}-y^3\right) d y

passes through

(0,2)

, then value of

(y(4))^2\left(4-(y(4))^2\right)

View answer and explanation

Correct option: A

Option A (16) is correct. The solution to the differential equation is found, and the given point is used to find the constant of integration. Finally, y(4) is calculated and substituted into the expression.

22 Hard

22 of 29

Application +4 | -1

Solution of the equation

x d y=\left(y+\frac{x f(y / x)}{f^{\prime}(y / x)}\right) d x

is -

View answer and explanation

Correct option: C

|f(y / x)|=c|x|, c>0

Option C (

|f(y / x)|=c|x|, c>0

) is correct because it represents the general solution to the given homogeneous differential equation.

23 Hard

23 of 29

Application +4 | -1

The equation of the curve satisfying the differential equation

{y_2}\left( {{x^2} + 1} \right) = 2x{y_1}

passing through the point (0,1) and having slope of tangent at

x = 0

as 3 is

View answer and explanation

Correct option: A

y = {x^3} + 3x + 1

Option A (

y = {x^3} + 3x + 1

) is correct because it satisfies the given differential equation and the initial conditions.

24 Hard

24 of 29

Application +4 | -1

The population of a city increases at the rate of

1 \%

per year. How many years will it take to double the population?

View answer and explanation

Correct option: D

30.1 years

Option D (30.1 years) is correct because it is the time it takes for the population to double at a 1% annual growth rate, calculated using exponential growth.

25 Hard

25 of 29

Application +4 | -1

y(y d x+x d y)=x^3(x d y-y d x): y(1)=1

Then

View answer and explanation

Correct option: B

2 y=x\left(x^2+y^2\right)

Option B (

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

) is correct because it is the solution to the given differential equation with the initial condition y(1)=1.

26 Hard

26 of 29

Application +4 | -1

y(x)

is the solution of the differential equation

(x \log x) \frac{d y}{d x}+y=2 x \log x(x \geq 1)

, then

y(e)=\ldots \ldots

View answer and explanation

Correct option: C

Option C (2) is the correct value of y(e), obtained by solving the differential equation and applying the given condition.

27 Hard

27 of 29

Analyzing +4 | -1

Given

\frac{d y}{d x}=\frac{y}{x}\left(1+x y^{2}(1+\ln x)\right)

. If

y(1)=3

, then the value of

\frac{y^{2}(3)}{9}

is:

View answer and explanation

Correct option: B

\frac{1}{43+27 \ln 3}

Option A (\

-\frac{1}{43+27 \ln 3}

) is correct because it is the value of \frac{{y²(3)}}{{9}} calculated by solving the given differential equation with the initial condition y(1)=3.

28 Hard

28 of 29

Analyzing +4 | -1

The solution of the differential equation

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

View answer and explanation

Correct option: A

\left| {\frac{x}{y}} \right| + \frac{{xy}}{{\left( {x - y} \right)}} = c

Option A (In

\left| {\frac{x}{y}} \right| + \frac{{xy}}{{\left( {x - y} \right)}} = c

) is correct because it represents the solution obtained by integrating the given differential equation.

29 Hard

29 of 29

Application +4 | -1

For the function

y={Bx}^2

to be the solution of differential equation

\left(\frac{d y}{d x}\right)^3-15 x^2 \frac{d y}{d x}-2 x y=0

, the value of

B

is_____ , given that

{B} \neq 0

View answer and explanation

Correct option: A

Option A (6) is correct because substituting the given condition y=2 when x=1 into the general solution yields the constant term c = 6.

Revision notes and key takeaways

Review DIFFERENTIAL EQUATIONS in your textbook or class notes, then use this set to check your understanding.

Key Takeaways

✓ This set contains 29 single-correct-answer MCQs on DIFFERENTIAL EQUATIONS.
✓ Try each question before opening its answer. Read the explanation where one is available.
✓ Sign in to save submitted answers. Revisit incorrect answers alongside your textbook or class notes.

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