The solution of the equation $\frac{d y}{d x}=\frac{x+y}{x-y}$ is

$c\left(x^2+y^2\right)^{1 / 2}+e^{\tan ^{-1}(y / x)}=0$

$c\left(x^2-y^2\right)=e^{\operatorname{tin}^{-1}(y / x)}$

The solution of $y' = 1 + x + {y^2} + x{y^2},y\left( 0 \right) = 0$ is

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

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

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

If $\frac{{dy}}{{dx}} + y = 2{e^{2x}}$, then $y$ is equal to

$\left( {1 - x} \right){e^{ - x}} + \frac{2}{3}{e^{2x}} + c$

${e^{ - x}} + \frac{2}{3}{e^{2x}} + c$

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JEE • MATHEMATICS • DIFFERENTIAL EQUATIONS

Differential equations practice · Medium

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121 Medium

121 of 452

Application +4 | -1

y = a{e^{mx}} + b{e^{ - mx}}

satisfies which of the following differential equations

View answer and explanation

Correct option: C

\frac{{{d^2}y}}{{d{x^2}}} - {m^2}y = 0

Option C (

\frac{{{d^2}y}}{{d{x^2}}} - {m^2}y = 0

) is correct because it is the differential equation that the given function satisfies after taking the second derivative.

122 Medium

122 of 452

Application +4 | -1

The solution of the differential equation

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

View answer and explanation

Correct option: B

{e^{ - y}} = {e^{ - x}} - {e^x} + c

Option B (

{e^{ - y}} = {e^{ - x}} - {e^x} + c

) is correct because it represents the solution of the given differential equation after separation of variables and integration.

123 Medium

123 of 452

Application +4 | -1

The solution of the equation

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

View answer and explanation

Correct option: B

c\left(x^2+y^2\right)^{1 / 2}=e^{\tan ^{-1}(y / x)}

Option B (

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

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

y=vx

124 Medium

124 of 452

Application +4 | -1

The solution of

y' = 1 + x + {y^2} + x{y^2},y\left( 0 \right) = 0

View answer and explanation

Correct option: D

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

Option D,

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

, is the correct solution because it satisfies the given differential equation and the initial condition

y(0) = 0

125 Medium

125 of 452

Application +4 | -1

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

is a solution of the differential equation

View answer and explanation

Correct option: D

{y_2} - {y_1} - 2y = 0

Option D (

{y_2} - {y_1} - 2y = 0

) is correct because it is the differential equation that is satisfied by the given function

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

126 Medium

126 of 452

Analyzing +4 | -1

The solution of the differential equation

{y_1}{y_3} = 3y_2^2

View answer and explanation

Correct option: A

x = {A_1}{y^2} + {A_2}y + {A_3}

Option A (

x = {A_1}{y^2} + {A_2}y + {A_3}

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

127 Medium

127 of 452

Application +4 | -1

The solution of

\frac{d y}{d x}=\frac{a x+b}{c y+d}

represents a parabola if

View answer and explanation

Correct option: C

a=0, c \neq 0

Option C (

a=0, c \neq 0

) is correct because it results in an equation where one variable is squared and the other is linear, which represents a parabola.

128 Medium

128 of 452

Application +4 | -1

A curve passing through

(2,3)

and satisfying the differential equation

\int_0^x t y(t) d t=x^2 y(x),(x>0)

View answer and explanation

Correct option: D

x y=6

Option D is correct because the curve xy=6 satisfies the given differential equation and passes through the point (2,3).

129 Medium

129 of 452

Analyzing +4 | -1

Let

x(t)=2 \sqrt{2} \cos t \sqrt{\sin 2 t}

and

y\left( t \right) = 2\sqrt 2 \sin t\sqrt {\sin 2t} ,\,\,\,t \notin \left( {0,\frac{\pi }{2}} \right).

Then

\frac{{1 + {{\left( {\frac{{dy}}{{dx}}} \right)}^2}}}{{\frac{{{d^2}y}}{{d{x^2}}}}}\,at\,\,t = \frac{\pi }{4}

is eqaul to

View answer and explanation

Correct option: D

\frac{{ - 2}}{3}

Option D (-2/3) is correct because after finding the first and second derivatives and substituting t = pi/4, the expression evaluates to -2/3.

130 Medium

130 of 452

Application +4 | -1

y = y\left( x \right)

and

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

then

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

equals

View answer and explanation

Correct option: A

\frac{1}{3}

Option A (

\frac{1}{3}

) is correct because it is the value of y at

x = \frac{\pi}{2}

after solving the differential equation and applying the initial condition.

131 Medium

131 of 452

Application +4 | -1

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

has the solution

View answer and explanation

Correct option: A

y+x^2+2 x+2=c e^x

Option A (

y+x^2+2 x+2=c e^x

) is correct because it is the solution obtained by solving the given first-order linear differential equation.

132 Medium

132 of 452

Application +4 | -1

Solution of differential equation

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

View answer and explanation

Correct option: C

y=x^2+a x

Option C (

y=x^2+a x

) is correct because it is the solution to the given differential equation after finding the integrating factor and integrating.

133 Medium

133 of 452

Application +4 | -1

Solution of the differential equation

\frac{d y}{d x}=\mathrm{y} \tan x-2 \sin \mathrm{x}

, is

View answer and explanation

Correct option: D

None of these

Option D (None of these) is correct because the solution to the differential equation does not match any of the given options.

134 Medium

134 of 452

Application +4 | -1

The solution of the equation

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

is [Where

A

is any arbitrary constant]

View answer and explanation

Correct option: B

y^3-x=A y

Option B (

y^3-x=A y

) is correct because it is the solution to the given differential equation after rearranging and solving as a linear equation in x.

135 Medium

135 of 452

Analyzing +4 | -1

The solution of differential equation

t = 1 + \left( {ty} \right)\frac{{dy}}{{dt}} + \frac{{{{\left( {ty} \right)}^2}}}{{2!}}{\left( {\frac{{dy}}{{dx}}} \right)^2} + ...\infty

View answer and explanation

Correct option: A

y = \pm \sqrt {{{(\log t)}^2} + c}

Option A (

y = \pm \sqrt {{{(\log t)}^2} + c}

) is correct because it is the solution obtained by solving the given differential equation after recognizing it as an exponential series.

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136 Medium

136 of 452

Analyzing +4 | -1

Solution of the differential equation

\frac{\sqrt{x} \,d x+\sqrt{y} \,d y}{\sqrt{x} \,d x-\sqrt{y} \,d y}=\sqrt{\frac{y^3}{x^3}}

is given by

View answer and explanation

Correct option: D

None of these

Option D (None of these) is correct because the provided options do not match the correct solution obtained by solving the differential equation.

137 Medium

137 of 452

Application +4 | -1

\frac{d y}{d x}+y \cot x=\cos x, y\left(\frac{\pi}{2}\right)=\frac{1}{2}

then

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

View answer and explanation

Correct option: D

\frac{1}{4}

Option D (

\frac{{1}}{{4}}

) is correct because it is the value of

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

obtained by solving the given linear differential equation with the initial condition.

138 Medium

138 of 452

Application +4 | -1

The differential equation of system of concentric circles with centre (1,2) is

View answer and explanation

Correct option: B

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

Option B is correct because the differential equation represents the family of concentric circles with center (1,2) obtained by differentiating the circle equation and eliminating the arbitrary constant.

139 Medium

139 of 452

Application +4 | -1

By substituting

y=v x

, the solution of the differential equation

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

, is

View answer and explanation

Correct option: B

\frac{y^2}{2 x^2}=\log x+C

Option B (

\frac{{y^2}}{{2 x^2}}=\log x+C

) is correct because it is the solution obtained by substituting y = vx into the given differential equation and solving the resulting separable equation.

140 Medium

140 of 452

Application +4 | -1

The D.E of

x y=a e^x+b e^{-x}+x^2

View answer and explanation

Correct option: B

x y_2+2 y_1=x y+2-x^2

Option B (

x y_2+2 y_1=x y+2-x^2

) is correct because it represents the differential equation obtained by differentiating the given equation twice and eliminating the arbitrary constants.

141 Medium

141 of 452

Application +4 | -1

The solution of

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

, is

View answer and explanation

Correct option: B

y{\sec ^2}x = \sec x + C

Option B (

y{\sec ^2}x = \sec x + C

) is correct because it is the solution to the given first-order linear differential equation after finding the integrating factor and integrating.

142 Medium

142 of 452

Application +4 | -1

For the differential equation

\frac{d y}{d x}+i y=2 \sin (x)

, with initial condition

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

. If

y(\pi)=\frac{a}{b}+c \pi i

, for co-prime integers

a

and

b

, find

a b c

. [

i=\sqrt{-1}

denotes the imaginary unit.]

View answer and explanation

Correct option: A

Option A (6) is correct because solving the given differential equation with the initial condition and substituting

x = \pi

yields

y(\pi) = -\frac{3}{2} - \pi i

, leading to

abc = 6

143 Medium

143 of 452

Application +4 | -1

The equation of the curve passing through

(3,9)

, which satisfies

d y / d x=x+1 / x^2

View answer and explanation

Correct option: C

6 x y=3 x^3+29 x-6

Option C (

6 x y=3 x^3+29 x-6

) is correct because it represents the curve obtained after integrating the given differential equation and applying the initial condition (3,9).

144 Medium

144 of 452

Application +4 | -1

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

, then

y

is equal to

View answer and explanation

Correct option: C

c{e^{ - x}} + \frac{2}{3}{e^{2x}}

Option C (

c{e^{ - x}} + \frac{2}{3}{e^{2x}}

) is correct because it is the solution obtained after finding the integrating factor and integrating the differential equation.

145 Medium

145 of 452

Application +4 | -1

A function

y=f(x)

satisfies

(x+1) \cdot f^{\prime}(x)-2\left(x^2+x\right) f(x)=\frac{e^{x^2}}{(x+1)}, \forall x>-1

.
If

f(0)=5

, then

f(x)

View answer and explanation

Correct option: B

\left(\frac{6 x+5}{x+1}\right) \cdot e^{x^2}

Option B (

\left(\frac{{6 x+5}}{{x+1}}\right) \cdot e^{x^2}

) is correct because it satisfies the given differential equation and the initial condition f(0)=5.

146 Medium

146 of 452

Application +4 | -1

The solution of the differential equation

\frac{{dy}}{{dx}} = \frac{{x\log {x^2} + x}}{{\sin y + y\cos y}}

, is

View answer and explanation

Correct option: A

y\sin y = {x^2}\log x + C

Option A (

y\sin y = {x^2}\log x + C

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

147 Medium

147 of 452

Analyzing +4 | -1

The differential equation whose solution is

{\left( {x - h} \right)^2} + {\left( {y - k} \right)^2} = {a^2}

is (

a

is a constant)

View answer and explanation

Correct option: B

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

Option B (

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

) is correct because it is the differential equation obtained by differentiating the given equation twice and eliminating the arbitrary constants h and k.

148 Medium

148 of 452

Application +4 | -1

The solution of the differential equation

\left(1-x y-x^5 y^5\right) d x-x^2\left(x^4 y^4+1\right) d y=0

is given by

View answer and explanation

Correct option: A

x=c e^{x y+\frac{1}{5} x^5 y^5}

Option A (

x=c e^{x y+\frac{1}{5} x^5 y^5}

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

149 Medium

149 of 452

Application +4 | -1

Solution of the differential equation

\left(y^3-2 x^2 y\right) d x+\left(2 x y^2-x^3\right) d y=0

View answer and explanation

Correct option: B

x y \sqrt{y^2-x^2}=c

Option B (

x y \sqrt{y^2-x^2}=c

) is correct because it is the solution obtained after solving the homogeneous differential equation using the substitution

y=vx

150 Medium

150 of 452

Application +4 | -1

Let

y=y(x)

be the solution of the differential equation

\frac{d y}{d x}+2 y=f(x)

, where

f(x)= \begin{cases}1, & x \in[0,1] \\ 0, & \text { otherwise }\end{cases}

y(0)=0

, then

y\left(\frac{3}{2}\right)

View answer and explanation

Correct option: A

\frac{e^2-1}{2 e^3}

Option A ((e^2-1)/(2e³)) is correct because solving the differential equation with the piecewise function and applying the initial condition yields y(3/2) = (e^2-1)/(2e³).

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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