The solution of differential equation $\frac{{dt}}{{dx}} = \frac{{t\left( {\frac{d}{{dx}}\left( {{\rm{g}}\left( x \right)} \right)} \right) - {t^2}}}{{{\rm{g}}\left( x \right)}}$ is

$t = \frac{{{\rm{g}}\left( x \right)}}{{x + c}}$

$t = \frac{{{\rm{g}}\left( x \right) + c}}{x}$

$t = \frac{{{\rm{g}}\left( x \right)}}{x} + c$

$t = {\rm{g}}\left( x \right) + x + c$

The solution of $(y+x+5) \mathrm{d} y=(y-x+$ 1) $\mathrm{d} x$ is

$\log \left((y+3)^2+(x+2)^2\right)+\tan ^{-1} \frac{y+3}{x+2}=C$

$\log \left((y+3)^2+(x-2)^2\right)+\tan ^{-1} \frac{y-3}{x-2}=C$

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JEE • MATHEMATICS • DIFFERENTIAL EQUATIONS

Differential equations practice · Medium

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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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151 Medium

151 of 452

Application +4 | -1

x d y=\left(2 y+2 x^4+x^2\right) d x, y(1)=0

then

y(e)=

View answer and explanation

Correct option: D

\mathrm{e}^4

Option D (

e^4

) is correct because it is the solution obtained after solving the given differential equation with the initial condition y(1)=0.

152 Medium

152 of 452

Application +4 | -1

Solution of differential equations

(x \cos x-\sin x) d x=\frac{x}{y} \sin x d y

View answer and explanation

Correct option: C

\left|\frac{\sin x}{x y}\right|=c

Option C (

\left|\frac{{\sin x}}{{x y}}\right|=c

) is correct because it represents the solution obtained after integrating the given differential equation and simplifying the result.

153 Medium

153 of 452

Application +4 | -1

Solution of equation

\frac{d y}{d x}=\frac{3 x-4 y-2}{3 x-4 y-3}

View answer and explanation

Correct option: B

x-y+c=\log (3 x-4 y+1)

Option B (

x-y+c=\log (3 x-4 y+1)

) is correct because it is the solution obtained by substituting

3x-4y=z

and integrating the resulting differential equation.

154 Medium

154 of 452

Application +4 | -1

The solution of differential equation

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

, is

View answer and explanation

Correct option: B

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

Option B (

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

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

155 Medium

155 of 452

Analyzing +4 | -1

Solution to the differential equation

\frac{x+\frac{x^3}{3 !}+\frac{x^3}{5 !}+\cdots}{1+\frac{x^2}{2 !}+\frac{x^4}{4 !}+\cdots}=\frac{d x-d y}{d x+d y}

View answer and explanation

Correct option: D

2 y e^{2 x}=c e^{2 x}-1

Option D (

2 y e^{2 x}=c e^{2 x}-1

) is correct because it is the solution obtained by solving the given differential equation using componendo and dividendo and integrating.

156 Medium

156 of 452

Application +4 | -1

The solution of the differential equation

y d x+\left(x+x^2 y\right) d y=0

View answer and explanation

Correct option: B

-\frac{1}{x y}+\log y=c

Option B (

-\frac{1}{x y}+\log y=c

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

157 Medium

157 of 452

Application +4 | -1

The solution curve of the differential equation

y \frac{d x}{d y}=x\left(\log _e x-\log _e y+1\right), x>0, y>0 \text { passing }

through the point

(e, 1)

View answer and explanation

Correct option: A

\left|\log _e \frac{x}{y}\right|=y

Option A (

\left|\log _e \frac{x}{y}\right|=y

) is correct because it involves solving the given differential equation and applying the initial condition to find the particular solution.

158 Medium

158 of 452

Application +4 | -1

The solution of differential equation

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

View answer and explanation

Correct option: A

y=e^{\frac{x}{y}}+C

Option A (

y=e^{\frac{x}{y}}+C

) is correct because it is the solution obtained after solving the given homogeneous differential equation by substitution and integration.

159 Medium

159 of 452

Application +4 | -1

The curve satisfying the differential equation

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

and passing through

(4,-2)

View answer and explanation

Correct option: B

y=-2 x

Option C,

y^3=-2 x

, is the correct curve obtained by solving the given differential equation and applying the given condition (4,-2).

160 Medium

160 of 452

Application +4 | -1

General solution of the differential equation

\left(\cos ^2 x\right) \frac{d y}{d x}+y=\tan x \,\,\,\left(0 \leq x < \frac{\pi}{2}\right)

View answer and explanation

Correct option: A

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

Option A (

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

) is correct because it is the solution obtained after solving the given linear differential equation.

161 Medium

161 of 452

Application +4 | -1

The solution of

y^5 x+y-x \frac{\mathrm{d} y}{\mathrm{~d} x}=0

View answer and explanation

Correct option: B

x^5 / 5+(1 / 4)(x / y)^4=C

Option B (

x^5 / 5+(1 / 4)(x / y)^4=C

) is correct because it is the solution obtained by multiplying the given differential equation by an appropriate integrating factor and then integrating.

162 Medium

162 of 452

Understanding +4 | -1

The order and degree of the differential equation

{\left( {1 + 3\frac{{dy}}{{dx}}} \right)^{\frac{2}{3}}} = 4\frac{{{d^3}y}}{{d{x^3}}}

are

View answer and explanation

Correct option: C

\left( {3,3} \right)

Option C (

3,3

) is correct because after raising both sides to the power of 3/2, the highest order derivative is

\frac{{{d^3}y}}{{d{x^3}}}

(order 3) and its power is 3 (degree 3).

163 Medium

163 of 452

Analyzing +4 | -1

The differential equation representing the family of ellipse having foci either on the

x

-axis or on the

y

-axis centre at the origin and passing through the point

(0,3)

View answer and explanation

Correct option: C

x y y^{\prime \prime}+x\left(y^{\prime}\right)^2-y y^{\prime}=0

Option C (

x y y^{\prime\prime}+x(y^{\prime})^2-y y^{\prime}=0

) is correct because it represents the differential equation of the family of ellipses centered at the origin, passing through (0,3), and having foci on either the x-axis or y-axis.

164 Medium

164 of 452

Application +4 | -1

A curve having the condition that the slope of tangent at some point is two times the slope of the straight line joining the same point to the origin of coordinates, is a/an

View answer and explanation

Correct option: C

Parabola

The correct answer is C (Parabola) because the given condition leads to a differential equation whose solution is of the form y = cx², which represents a parabola.

165 Medium

165 of 452

Application +4 | -1

General solution of the equation

\left(1+x^2\right) \frac{d y}{d x}+x y=x^3 y^3

View answer and explanation

Correct option: B

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

Option B (

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

) is the correct general solution obtained by solving the Bernoulli differential equation.

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166 Medium

166 of 452

Application +4 | -1

The solution of

\frac{{d\upsilon }}{{dt}} + \frac{k}{m}\upsilon = - g

View answer and explanation

Correct option: A

\upsilon = c{e^{ - \frac{k}{m}t}} - \frac{{mg}}{k}

Option A (

\upsilon = c{e^{ - \frac{k}{m}t}} - \frac{{mg}}{k}

) is correct because it is the solution to the given first-order linear differential equation.

167 Medium

167 of 452

Analyzing +4 | -1

y = {\left( {x + \sqrt {1 + x} } \right)^n}

, then

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

View answer and explanation

Correct option: A

{n^2}y

Option A (

{n^2}y

) is correct because after differentiating the given equation twice and simplifying, we obtain the expression equal to

{n^2}y

168 Medium

168 of 452

Analyzing +4 | -1

x=\sec \theta-\cos \theta

and

y=\sec ^n \theta-\cos ^n \theta

, then

\left(x^2+4\right)\left(\frac{d y}{d x}\right)^2

is equal to

View answer and explanation

Correct option: C

n^2\left(y^2+4\right)

The correct answer is C (

n^2\left(y^2+4\right)

) because it is the simplified expression obtained after differentiating

x

and

y

with respect to

\theta

and substituting into the given equation.

169 Medium

169 of 452

Application +4 | -1

The solution of the differential equation

\frac{{dy}}{{dx}} = \frac{{x - 2y + 1}}{{2x - 4y}}

View answer and explanation

Correct option: A

{\left( {x - 2y} \right)^2} + 2x = c

Option A (

{\left( {x - 2y} \right)^2} + 2x = c

) is correct because it is the solution to the differential equation obtained by using a suitable substitution.

170 Medium

170 of 452

Application +4 | -1

The solution of differential equation

\frac{{dt}}{{dx}} = \frac{{t\left( {\frac{d}{{dx}}\left( {\mathrm{g}\left( x \right)} \right)} \right) - {t^2}}}{{\mathrm{g}\left( x \right)}}

View answer and explanation

Correct option: C

t = \frac{{\mathrm{g}\left( x \right)}}{{x + c}}

Option D (

t = \frac{{\mathrm{g}\left( x \right)}}}{{x + c}}

) is correct because it is the solution to the given differential equation after applying a suitable substitution and integrating factor.

171 Medium

171 of 452

Application +4 | -1

The solution of differential equation

\left(x y^2-e^{x^{\frac{1}{3}}}\right) d x-x^2 y d y=0

View answer and explanation

Correct option: B

3 y^2=2 x^2 e^{x^{\frac{1}{3}}}+C x^2

Option B (

3 y^2=2 x^2 e^{x^{\frac{1}{3}}}+C x^2

) is correct as it represents the solution to the given differential equation after using an integrating factor.

172 Medium

172 of 452

Application +4 | -1

The solution of

2\left( {y + 3} \right) - xy\frac{{dy}}{{dx}} = 0

with

y = - 2

, where

x = 1

, is

View answer and explanation

Correct option: D

{x^2}{\left( {y + 3} \right)^3} = {e^{y + 2}}

The correct answer is D (

{x^2}{\left( {y + 3} \right)^3} = {e^{y + 2}}

) because it is the solution to the given differential equation

2\left( {y + 3} \right) - xy\frac{{dy}}{{dx}} = 0

with the initial condition

y = -2

when

x = 1

173 Medium

173 of 452

Application +4 | -1

By a suitable substitution, the equation

y^3 \frac{d y}{d x}+x+y^2=0

can be transformed to

View answer and explanation

Correct option: B

Homogeneous

Option B (Homogeneous) is correct because the given equation can be transformed into a homogeneous differential equation by using the substitution

z = x + y^2

174 Medium

174 of 452

Application +4 | -1

The general solution of

ydx - xdy - 3{x^2}{y^2}{e^{{x^3}}}dx = 0

, is equal to

View answer and explanation

Correct option: A

\frac{x}{y} = {e^{{x^3}}} + C

Option A (

\frac{x}{y} = {e^{{x^3}}} + C

) is correct because it is the general solution obtained by rearranging the given differential equation and integrating.

175 Medium

175 of 452

Application +4 | -1

The solution curve, of the differential equation

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

, passing through the point

(0,1)

is a conic, whose vertex lies on the line:

View answer and explanation

Correct option: A

2 x+3 y=9

Option A (

2 x+3 y=9

) is correct because the vertex of the conic (parabola) lies on this line, after solving the differential equation and applying the given point.

176 Medium

176 of 452

Application +4 | -1

The equation of curve through point (1,0) which satisfies the differential equation

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

View answer and explanation

Correct option: B

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

Option B (

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

) is correct because it is the solution to the differential equation that passes through the point (1,0).

177 Medium

177 of 452

Application +4 | -1

General solution of the equation

\frac{d y}{d x}+\frac{1}{x}=\frac{e^y}{x^2}

View answer and explanation

Correct option: A

2 x=e^y\left(2 c x^2+1\right)

Option A (

2 x=e^y(2 c x^2+1)

) is the correct general solution obtained by solving the differential equation.

178 Medium

178 of 452

Application +4 | -1

The solution of

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

View answer and explanation

Correct option: B

y = \left( {x + 1} \right)\log \left| {x + 1} \right| - x + 3

Option B,

y = \left( {x + 1} \right)\log \left| {x + 1} \right| - x + 3

is the correct solution obtained by integrating the given differential equation and applying the initial condition.

179 Medium

179 of 452

Application +4 | -1

The solution of

(y+x+5) \mathrm{d} y=(y-x+

\mathrm{d} x

View answer and explanation

Correct option: C

\log \left((y+3)^2+(x+2)^2\right)+2 \tan ^{-1} \frac{y+3}{x+2}=C

The correct answer is C (

\log \left((y+3)^2+(x+2)^2\right)+2 \tan ^{-1} \frac{{y+3}}{{x+2}}=C

) because it is the solution to the homogeneous differential equation after substitution and integration.

180 Medium

180 of 452

Application +4 | -1

y_1(x)

is a solution of the differential equation

\frac{d y}{d x}-f(x) y=0

, then a solution of the differential equation

\frac{d y}{d x}+f(x) y=r(x)

can be

y=

View answer and explanation

Correct option: A

\frac{1}{y_1(x)} \int r(x) y_1(x) d x

Option A (

\frac{{1}}{{y_1(x)}} \int r(x) y_1(x) dx

) is correct because it represents the solution obtained using the integrating factor method for a first-order linear differential equation.

Revision notes and key takeaways

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

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