JEE Main

Differential Equations

Mathematics

(Past Years Questions)

MCQ (Single Correct Answer)

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If y = y(x) is the solution curve of the differential equation $${x^2}dy + \left( {y - {1 \over x}} \right)dx = 0$$ ; x ...
JEE Main 2021 (Online) 1st September Evening Shift
If $$y{{dy} \over {dx}} = x\left[ {{{{y^2}} \over {{x^2}}} + {{\phi \left( {{{{y^2}} \over {{x^2}}}} \right)} \over {\ph...
JEE Main 2021 (Online) 31st August Evening Shift
If $${{dy} \over {dx}} = {{{2^x}y + {2^y}{{.2}^x}} \over {{2^x} + {2^{x + y}}{{\log }_e}2}}$$, y(0) = 0, then for y = 1,...
JEE Main 2021 (Online) 31st August Evening Shift
If $${{dy} \over {dx}} = {{{2^{x + y}} - {2^x}} \over {{2^y}}}$$, y(0) = 1, then y(1) is equal to :
JEE Main 2021 (Online) 31st August Morning Shift
If the solution curve of the differential equation (2x $$-$$ 10y3)dy + ydx = 0, passes through the points (0, 1) and (2,...
JEE Main 2021 (Online) 27th August Evening Shift
A differential equation representing the family of parabolas with axis parallel to y-axis and whose length of latus rect...
JEE Main 2021 (Online) 27th August Evening Shift
Let us consider a curve, y = f(x) passing through the point ($$-$$2, 2) and the slope of the tangent to the curve at any...
JEE Main 2021 (Online) 27th August Morning Shift
Let y = y(x) be the solution of the differential equation $${{dy} \over {dx}} = 2(y + 2\sin x - 5)x - 2\cos x$$ such tha...
JEE Main 2021 (Online) 27th August Morning Shift
Let y(x) be the solution of the differential equation 2x2 dy + (ey $$-$$ 2x)dx = 0, x > 0. If y(e) = 1, then y(1) is ...
JEE Main 2021 (Online) 26th August Evening Shift
Let y = y(x) be a solution curve of the differential equation $$(y + 1){\tan ^2}x\,dx + \tan x\,dy + y\,dx = 0$$, $$x \i...
JEE Main 2021 (Online) 26th August Morning Shift
Let y = y(x) be solution of the differential equation $${\log _{}}\left( {{{dy} \over {dx}}} \right) = 3x + 4y$$, with y...
JEE Main 2021 (Online) 27th July Morning Shift
Let y = y(x) be the solution of the differential equation xdy = (y + x3 cosx)dx with y($$\pi$$) = 0, then $$y\left( {{\p...
JEE Main 2021 (Online) 25th July Evening Shift
Let y = y(x) be the solution of the differential equation (x $$-$$ x3)dy = (y + yx2 $$-$$ 3x4)dx, x > 2. If y(3) = 3,...
JEE Main 2021 (Online) 27th July Evening Shift
Let y = y(x) be the solution of the differential equation $${{dy} \over {dx}} = 1 + x{e^{y - x}}, - \sqrt 2 < x <...
JEE Main 2021 (Online) 25th July Morning Shift
Let y = y(x) be the solution of the differential equation $$\cos e{c^2}xdy + 2dx = (1 + y\cos 2x)\cos e{c^2}xdx$$, with ...
JEE Main 2021 (Online) 22th July Evening Shift
Let y = y(x) be the solution of the differential equation $${e^x}\sqrt {1 - {y^2}} dx + \left( {{y \over x}} \right)dy =...
JEE Main 2021 (Online) 20th July Morning Shift
Let y = y(x) be the solution of the differential equation $$x\tan \left( {{y \over x}} \right)dy = \left( {y\tan \left( ...
JEE Main 2021 (Online) 20th July Morning Shift
Let y = y(x) be the solution of the differential equation $${{dy} \over {dx}} = (y + 1)\left( {(y + 1){e^{{x^2}/2}} - x}...
JEE Main 2021 (Online) 18th March Evening Shift
The differential equation satisfied by the system of parabolas y2 = 4a(x + a) is :
JEE Main 2021 (Online) 18th March Morning Shift
Let y = y(x) be the solution of the differential equation $$\cos x(3\sin x + \cos x + 3)dy = (1 + y\sin x(3\sin x + \cos...
JEE Main 2021 (Online) 17th March Evening Shift
If the curve y = y(x) is the solution of the differential equation $$2({x^2} + {x^{5/4}})dy - y(x + {x^{1/4}})dx = {2^{9...
JEE Main 2021 (Online) 17th March Evening Shift
Which of the following is true for y(x) that satisfies the differential equation $${{dy} \over {dx}}$$ = xy $$-$$ 1 + x ...
JEE Main 2021 (Online) 17th March Morning Shift
Let C1 be the curve obtained by the solution of differential equation $$2xy{{dy} \over {dx}} = {y^2} - {x^2},x > 0$$....
JEE Main 2021 (Online) 16th March Evening Shift
If y = y(x) is the solution of the differential equation $${{dy} \over {dx}}$$ + (tan x) y = sin x, $$0 \le x \le {\pi ...
JEE Main 2021 (Online) 16th March Evening Shift
If y = y(x) is the solution of the differential equation, $${{dy} \over {dx}} + 2y\tan x = \sin x,y\left( {{\pi \over 3...
JEE Main 2021 (Online) 16th March Morning Shift
Let $$f(x) = \int\limits_0^x {{e^t}f(t)dt + {e^x}} $$ be a differentiable function for all x$$\in$$R. Then f(x) equals :
JEE Main 2021 (Online) 26th February Evening Shift
The rate of growth of bacteria in a culture is proportional to the number of bacteria present and the bacteria count is ...
JEE Main 2021 (Online) 26th February Morning Shift
If a curve passes through the origin and the slope of the tangent to it at any point (x, y) is $${{{x^2} - 4x + y + 8} \...
JEE Main 2021 (Online) 25th February Morning Slot
The population P = P(t) at time 't' of a certain species follows the differential equation $${{dP} \over {dt}}$$ = 0.5P ...
JEE Main 2021 (Online) 24th February Morning Slot
If $$y = \left( {{2 \over \pi }x - 1} \right) cosec\,x$$ is the solution of the differential equation, $${{dy} \over {d...
JEE Main 2020 (Online) 6th September Evening Slot
The general solution of the differential equation $$\sqrt {1 + {x^2} + {y^2} + {x^2}{y^2}} $$ + xy$${{dy} \over {dx}}$$ ...
JEE Main 2020 (Online) 6th September Morning Slot
Let y = y(x) be the solution of the differential equation cosx$${{dy} \over {dx}}$$ + 2ysinx = sin2x, x $$ \in $$ $$\lef...
JEE Main 2020 (Online) 5th September Evening Slot
If y = y(x) is the solution of the differential equation $${{5 + {e^x}} \over {2 + y}}.{{dy} \over {dx}} + {e^x} = 0$$ ...
JEE Main 2020 (Online) 5th September Morning Slot
The solution of the differential equation $${{dy} \over {dx}} - {{y + 3x} \over {{{\log }_e}\left( {y + 3x} \right)}} +...
JEE Main 2020 (Online) 4th September Evening Slot
Let f be a twice differentiable function on (1, 6). If f(2) = 8, f’(2) = 5, f’(x) $$ \ge $$ 1 and f''(x) $$ \ge $$ 4, fo...
JEE Main 2020 (Online) 4th September Morning Slot
Let y = y(x) be the solution of the differential equation, xy'- y = x2(xcosx + sinx), x > 0. if y ($$\pi $$) then $$y...
JEE Main 2020 (Online) 4th September Morning Slot
The solution curve of the differential equation, (1 + e-x)(1 + y2)$${{dy} \over {dx}}$$ = y2, which passes through the p...
JEE Main 2020 (Online) 3rd September Morning Slot
If a curve y = f(x), passing through the point (1, 2), is the solution of the differential equation, 2x2dy= (2xy + y2)dx...
JEE Main 2020 (Online) 2nd September Evening Slot
If $${{dy} \over {dx}} = {{xy} \over {{x^2} + {y^2}}}$$; y(1) = 1; then a value of x satisfying y(x) = e is :
JEE Main 2020 (Online) 9th January Evening Slot
The differential equation of the family of curves, x2 = 4b(y + b), b $$ \in $$ R, is
JEE Main 2020 (Online) 8th January Evening Slot
Let y = y(x) be a solution of the differential equation, $$\sqrt {1 - {x^2}} {{dy} \over {dx}} + \sqrt {1 - {y^2}} = 0$...
JEE Main 2020 (Online) 8th January Morning Slot
Let y = y(x) be the solution curve of the differential equation, $$\left( {{y^2} - x} \right){{dy} \over {dx}} = 1$$, sa...
JEE Main 2020 (Online) 7th January Evening Slot
If y = y(x) is the solution of the differential equation, $${e^y}\left( {{{dy} \over {dx}} - 1} \right) = {e^x}$$ such t...
JEE Main 2020 (Online) 7th January Morning Slot
Let xk + yk = ak, (a, k > 0 ) and $${{dy} \over {dx}} + {\left( {{y \over x}} \right)^{{1 \over 3}}} = 0$$, then k is...
JEE Main 2020 (Online) 7th January Morning Slot
The general solution of the differential equation (y2 – x3)dx – xydy = 0 (x $$ \ne $$ 0) is : (where c is a constant of ...
JEE Main 2019 (Online) 12th April Evening Slot
Consider the differential equation, $${y^2}dx + \left( {x - {1 \over y}} \right)dy = 0$$, If value of y is 1 when x = 1,...
JEE Main 2019 (Online) 12th April Morning Slot
Let y = y(x) be the solution of the differential equation, $${{dy} \over {dx}} + y\tan x = 2x + {x^2}\tan x$$, $$x \in \...
JEE Main 2019 (Online) 10th April Evening Slot
If y = y(x) is the solution of the differential equation $${{dy} \over {dx}} = \left( {\tan x - y} \right){\sec ^2}x$$,...
JEE Main 2019 (Online) 10th April Morning Slot
If $$\cos x{{dy} \over {dx}} - y\sin x = 6x$$, (0 < x < $${\pi \over 2}$$) and $$y\left( {{\pi \over 3}} \right)...
JEE Main 2019 (Online) 9th April Evening Slot
The solution of the differential equation $$x{{dy} \over {dx}} + 2y$$ = x2 (x $$ \ne $$ 0) with y(1) = 1, is...
JEE Main 2019 (Online) 9th April Morning Slot
Let y = y(x) be the solution of the differential equation, $${({x^2} + 1)^2}{{dy} \over {dx}} + 2x({x^2} + 1)y = 1$$ suc...
JEE Main 2019 (Online) 8th April Morning Slot
If a curve passes through the point (1, –2) and has slope of the tangent at any point (x, y) on it as $${{{x^2} - 2y} \o...
JEE Main 2019 (Online) 12th January Evening Slot
Let y = y(x) be the solution of the differential equation, x$${{dy} \over {dx}}$$ + y = x loge x, (x > 1). If 2y(2)...
JEE Main 2019 (Online) 12th January Morning Slot
The solution of the differential equation, $${{dy} \over {dx}}$$ = (x – y)2, when y(1) = 1, is :
JEE Main 2019 (Online) 11th January Evening Slot
If  xloge(logex) $$-$$ x2 + y2 = 4(y > 0), then $${{dy} \over {dx}}$$ at x = e is equal to : ...
JEE Main 2019 (Online) 11th January Morning Slot
A helicopter is flying along the curve given by y – x3/2 = 7, (x $$ \ge $$ 0). A soldier positioned at the point $$\left...
JEE Main 2019 (Online) 10th January Evening Slot
Let f be a differentiable function such that f '(x) = 7 - $${3 \over 4}{{f\left( x \right)} \over x},$$ (x > 0) and f...
JEE Main 2019 (Online) 10th January Evening Slot
The curve amongst the family of curves represented by the differential equation, (x2 – y2)dx + 2xy dy = 0 which passes t...
JEE Main 2019 (Online) 10th January Evening Slot
Let f : [0,1] $$ \to $$ R be such that f(xy) = f(x).f(y), for all x, y $$ \in $$ [0, 1], and f(0) $$ \ne $$ 0. If ...
JEE Main 2019 (Online) 9th January Evening Slot
The maximum volume (in cu.m) of the right circular cone having slant height 3 m is :
JEE Main 2019 (Online) 9th January Morning Slot
If y = y(x) is the solution of the differential equation, x$$dy \over dx$$ + 2y = x2, satisfying y(1) = 1, then y($$1\o...
JEE Main 2019 (Online) 9th January Morning Slot
The differential equation representing the family of ellipse having foci eith on the x-axis or on the $$y$$-axis, center...
JEE Main 2018 (Online) 16th April Morning Slot
The curve satifying the differeial equation, (x2 $$-$$ y2) dx + 2xydy = 0 and passing through the point (1, 1) is : ...
JEE Main 2018 (Online) 15th April Evening Slot
Let y = y(x) be the solution of the differential equation $${{dy} \over {dx}} + 2y = f\left( x \right),$$ where $$f\left...
JEE Main 2018 (Online) 15th April Morning Slot
Let y = y(x) be the solution of the differential equation $$\sin x{{dy} \over {dx}} + y\cos x = 4x$$, $$x \in \left( {0,...
JEE Main 2018 (Offline)
Let S = { t $$ \in R:f(x) = \left| {x - \pi } \right|.\left( {{e^{\left| x \right|}} - 1} \right)$$$$\sin \left| x \righ...
JEE Main 2018 (Offline)
Let $$f\left( x \right) = {x^2} + {1 \over {{x^2}}}$$ and $$g\left( x \right) = x - {1 \over x}$$, $$x \in R - \left\{ {...
JEE Main 2018 (Offline)
If the curves y2 = 6x, 9x2 + by2 = 16 intersect each other at right angles, then the value of b is :...
JEE Main 2018 (Offline)
The curve satisfying the differential equation, ydx $$-$$(x + 3y2)dy = 0 and passing through the point (1, 1), also pas...
JEE Main 2017 (Online) 8th April Morning Slot
If $$\left( {2 + \sin x} \right){{dy} \over {dx}} + \left( {y + 1} \right)\cos x = 0$$ and y(0) = 1, then $$y\left( {{\p...
JEE Main 2017 (Offline)
The normal to the curve y(x – 2)(x – 3) = x + 6 at the point where the curve intersects the y-axis passes through the po...
JEE Main 2017 (Offline)
Twenty meters of wire is available for fencing off a flower-bed in the form of a circular sector. Then the maximum area ...
JEE Main 2017 (Offline)
The solution of the differential equation $${{dy} \over {dx}}\, + \,{y \over 2}\,\sec x = {{\tan x} \over {2y}},\,\,$$ ...
JEE Main 2016 (Online) 10th April Morning Slot
If   f(x) is a differentiable function in the interval (0, $$\infty $$) such that f (1) = 1 and $$\mathop {...
JEE Main 2016 (Online) 9th April Morning Slot
If a curve $$y=f(x)$$ passes through the point $$(1,-1)$$ and satisfies the differential equation, $$y(1+xy) dx=x$$ $$dy...
JEE Main 2016 (Offline)
Let $$y(x)$$ be the solution of the differential equation $$\left( {x\,\log x} \right){{dy} \over {dx}} + y = 2x\,\log ...
JEE Main 2015 (Offline)
Let the population of rabbits surviving at time $$t$$ be governed by the differential equation $${{dp\left( t \right)} ...
JEE Main 2014 (Offline)
At present, a firm is manufacturing $$2000$$ items. It is estimated that the rate of change of production P w.r.t. add...
JEE Main 2013 (Offline)
The population $$p$$ $$(t)$$ at time $$t$$ of a certain mouse species satisfies the differential equation $${{dp\left( t...
AIEEE 2012
If $${{dy} \over {dx}} = y + 3 > 0\,\,$$ and $$y(0)=2,$$ then $$y\left( {\ln 2} \right)$$ is equal to :
AIEEE 2011
Let $$I$$ be the purchase value of an equipment and $$V(t)$$ be the value after it has been used for $$t$$ years. The va...
AIEEE 2011
Solution of the differential equation $$\cos x\,dy = y\left( {\sin x - y} \right)dx,\,\,0 < x{\pi \over 2}$$ is ...
AIEEE 2010
The differential equation which represents the family of curves $$y = {c_1}{e^{{c_2}x}},$$ where $${c_1}$$ , and $${c_2}...
AIEEE 2009
The solution of the differential equation $${{dy} \over {dx}} = {{x + y} \over x}$$ satisfying the condition $$y(1)=1$$...
AIEEE 2008
The differential equation of all circles passing through the origin and having their centres on the $$x$$-axis is
AIEEE 2007
The differential equation whose solution is $$A{x^2} + B{y^2} = 1$$ where $$A$$ and $$B$$ are arbitrary constants is of...
AIEEE 2006
If $$x{{dy} \over {dx}} = y\left( {\log y - \log x + 1} \right),$$ then the solution of the equation is
AIEEE 2005
The differential equation representing the family of curves $${y^2} = 2c\left( {x + \sqrt c } \right),$$ where $$c>0,...
AIEEE 2005
Solution of the differential equation $$ydx + \left( {x + {x^2}y} \right)dy = 0$$ is
AIEEE 2004
The differential equation for the family of circle $${x^2} + {y^2} - 2ay = 0,$$ where a is an arbitrary constant is
AIEEE 2004
The degree and order of the differential equation of the family of all parabolas whose axis is $$x$$-axis, are respectiv...
AIEEE 2003
The solution of the differential equation $$\left( {1 + {y^2}} \right) + \left( {x - {e^{{{\tan }^{ - 1}}y}}} \right){{...
AIEEE 2003
The order and degree of the differential equation $$\,{\left( {1 + 3{{dy} \over {dx}}} \right)^{2/3}} = 4{{{d^3}y} \ove...
AIEEE 2002
The solution of the equation $$\,{{{d^2}y} \over {d{x^2}}} = {e^{ - 2x}}$$
AIEEE 2002

Numerical

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If $${y^{1/4}} + {y^{ - 1/4}} = 2x$$, and $$({x^2} - 1){{{d^2}y} \over {d{x^2}}} + \alpha x{{dy} \over {dx}} + \beta y =...
JEE Main 2021 (Online) 27th August Morning Shift
If $$y = y(x),y \in \left[ {0,{\pi \over 2}} \right)$$ is the solution of the differential equation $$\sec y{{dy} \over...
JEE Main 2021 (Online) 27th July Morning Shift
Let $$F:[3,5] \to R$$ be a twice differentiable function on (3, 5) such that $$F(x) = {e^{ - x}}\int\limits_3^x {(3{t^2}...
JEE Main 2021 (Online) 27th July Morning Shift
Let y = y(x) be the solution of the differential equation dy = e$$\alpha$$x + y dx; $$\alpha$$ $$\in$$ N. If y(loge2) = ...
JEE Main 2021 (Online) 27th July Evening Shift
Let y = y(x) be solution of the following differential equation $${e^y}{{dy} \over {dx}} - 2{e^y}\sin x + \sin x{\cos ^2...
JEE Main 2021 (Online) 25th July Morning Shift
Let y = y(x) be the solution of the differential equation $$\left( {(x + 2){e^{\left( {{{y + 1} \over {x + 2}}} \right)}...
JEE Main 2021 (Online) 22th July Evening Shift
Let a curve y = y(x) be given by the solution of the differential equation $$\cos \left( {{1 \over 2}{{\cos }^{ - 1}}({e...
JEE Main 2021 (Online) 20th July Evening Shift
Let y = y(x) be the solution of the differential equation xdy $$-$$ ydx = $$\sqrt {({x^2} - {y^2})} dx$$, x $$ \ge $$ 1,...
JEE Main 2021 (Online) 18th March Evening Shift
If y = y(x) is the solution of the equation $${e^{\sin y}}\cos y{{dy} \over {dx}} + {e^{\sin y}}\cos x = \cos x$$, y(0) ...
JEE Main 2021 (Online) 26th February Morning Shift
The difference between degree and order of a differential equation that represents the family of curves given by $${y^2}...
JEE Main 2021 (Online) 26th February Morning Shift
If the curve, y = y(x) represented by the solution of the differential equation (2xy2 $$-$$ y)dx + xdy = 0, passes throu...
JEE Main 2021 (Online) 25th February Evening Shift
If for x $$ \ge $$ 0, y = y(x) is the solution of the differential equation (x + 1)dy = ((x + 1)2 + y – 3)dx, y(2) = 0, ...
JEE Main 2020 (Online) 9th January Morning Slot

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