## Differential Equations

### Mathematics

#### (Past Years Questions)

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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$$such tha... JEE Main 2021 (Online) 27th August Morning Shift Let y(x) be the solution of the differential equation 2x2 dy + (ey$$-$$2x)dx = 0, x &gt; 0. If y(e) = 1, then y(1) is ... JEE Main 2021 (Online) 26th August Evening Shift Let y = y(x) be a solution curve of the differential equation$$(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 $$y\left( {{\p... JEE Main 2021 (Online) 25th July Evening Shift Let y = y(x) be the solution of the differential equation (x$$-$$x3)dy = (y + yx2$$-$$3x4)dx, x &gt; 2. If y(3) = 3,... JEE Main 2021 (Online) 27th July Evening Shift Let y = y(x) be the solution of the differential equation$${{dy} \over {dx}} = 1 + x{e^{y - x}}, - \sqrt 2 &lt; x &lt;...
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 &gt; 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 $$4, fo... JEE Main 2020 (Online) 4th September Morning Slot Let y = y(x) be the solution of the differential equation, xy'- y = x2(xcosx + sinx), x &gt; 0. if y ($$\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}$$such t... JEE Main 2020 (Online) 7th January Morning Slot Let xk + yk = ak, (a, k &gt; 0 ) and$${{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 &lt; x &lt; $${\pi \over 2}$$) 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 &gt; 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 &nbsp;xloge(logex) $$-$$ x2 + y2 = 4(y &gt; 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},$$(x &gt; 0) and f... JEE Main 2019 (Online) 10th January Evening Slot The curve amongst the family of curves represented by the differential equation, (x2 – y2)dx + 2xy dy = 0 which passes t... JEE Main 2019 (Online) 10th January Evening Slot Let f : [0,1]$$ \to $$&nbsp;R be such that f(xy) = f(x).f(y), for all x, y$$ \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}},\,\,$$... JEE Main 2016 (Online) 10th April Morning Slot If&nbsp;&nbsp; f(x) is a differentiable function in the interval (0,$$\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 $${{dy} \over {dx}} = y + 3 &gt; 0\,\,$$ 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 $$\cos x\,dy = y\left( {\sin x - y} \right)dx,\,\,0 &lt; x{\pi \over 2}$$ 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&gt;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