JEE Main
Mathematics
Differential Equations
Previous Years Questions

Let $\alpha$ be a non-zero real number. Suppose $f: \mathbf{R} \rightarrow \mathbf{R}$ is a differentiable function such that $f(0)=2$ and $\lim\limit... Let$y=y(x)$be the solution of the differential equation$\frac{\mathrm{d} y}{\mathrm{~d} x}=2 x(x+y)^3-x(x+y)-1, y(0)=1$. Then,$\left(\frac{1}{\sq...
The temperature $$T(t)$$ of a body at time $$t=0$$ is $$160^{\circ} \mathrm{F}$$ and it decreases continuously as per the differential equation $$\fra... Let$$y=y(x)$$be the solution of the differential equation$$\frac{d y}{d x}=\frac{(\tan x)+y}{\sin x(\sec x-\sin x \tan x)}, x \in\left(0, \frac{\pi...
The solution curve of the differential equation $$y \frac{d x}{d y}=x\left(\log _e x-\log _e y+1\right), x>0, y>0$$ passing through the point $$(e, 1... Let$$y=y(x)$$be the solution of the differential equation$$\sec x \mathrm{~d} y+\{2(1-x) \tan x+x(2-x)\} \mathrm{d} x=0$$such that$$y(0)=2$$. The... If$$\sin \left(\frac{y}{x}\right)=\log _e|x|+\frac{\alpha}{2}$$is the solution of the differential equation$$x \cos \left(\frac{y}{x}\right) \frac{...
A function $$y=f(x)$$ satisfies $$f(x) \sin 2 x+\sin x-\left(1+\cos ^2 x\right) f^{\prime}(x)=0$$ with condition $$f(0)=0$$. Then, $$f\left(\frac{\pi}... If$$y=y(x)$$is the solution curve of the differential equation$$\left(x^2-4\right) \mathrm{d} y-\left(y^2-3 y\right) \mathrm{d} x=0, x>2, y(4)=\fra...
Let $x=x(\mathrm{t})$ and $y=y(\mathrm{t})$ be solutions of the differential equations $\frac{\mathrm{d} x}{\mathrm{dt}}+\mathrm{a} x=0$ and $\frac{\m... Let$x=x(y)$be the solution of the differential equation$2(y+2) \log _{e}(y+2) d x+\left(x+4-2 \log _{e}(y+2)\right) d y=0, y>-1$with$x\left(e^{...
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 $$y_{1}(0)=0... Let$$y=y(x), y > 0$$, be a solution curve of the differential equation$$\left(1+x^{2}\right) \mathrm{d} y=y(x-y) \mathrm{d} x$$. If$$y(0)=1$$and ... Let$$y=y(x)$$be the solution of the differential equation$$\frac{d y}{d x}+\frac{5}{x\left(x^{5}+1\right)} y=\frac{\left(x^{5}+1\right)^{2}}{x^{7}}...
Let $$y=y(x)$$ be a solution curve of the differential equation. $$\left(1-x^{2} y^{2}\right) d x=y d x+x d y$$. If the line $$x=1$$ intersects the cu...
Let $$f$$ be a differentiable function such that $${x^2}f(x) - x = 4\int\limits_0^x {tf(t)dt}$$, $$f(1) = {2 \over 3}$$. Then $$18f(3)$$ is equal to ...
If the solution curve $$f(x, y)=0$$ of the differential equation $$\left(1+\log _{e} x\right) \frac{d x}{d y}-x \log _{e} x=e^{y}, x > 0$$, passes t...
Let $$\alpha x=\exp \left(x^{\beta} y^{\gamma}\right)$$ be the solution of the differential equation $$2 x^{2} y \mathrm{~d} y-\left(1-x y^{2}\right) ... The area enclosed by the closed curve$$\mathrm{C}$$given by the differential equation$$\frac{d y}{d x}+\frac{x+a}{y-2}=0, y(1)=0$$is$$4 \pi$$. Le... If$$y=y(x)$$is the solution curve of the differential equation$$\frac{d y}{d x}+y \tan x=x \sec x, 0 \leq x \leq \frac{\pi}{3}, y(0)=1$$, then$$y\...
Let $y=y(x)$ be the solution of the differential equation $\left(3 y^{2}-5 x^{2}\right) y \mathrm{~d} x+2 x\left(x^{2}-y^{2}\right) \mathrm{d} y=0$ ...
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 :...
The solution of the differential equation $\frac{d y}{d x}=-\left(\frac{x^2+3 y^2}{3 x^2+y^2}\right), y(1)=0$ is :
Let the solution curve $$y=y(x)$$ of the differential equation $$\frac{\mathrm{d} y}{\mathrm{~d} x}-\frac{3 x^{5} \tan ^{-1}\left(x^{3}\right)}{\left... Let$$y=y(x)$$be the solution of the differential equation$$x{\log _e}x{{dy} \over {dx}} + y = {x^2}{\log _e}x,(x > 1)$$. If$$y(2) = 2$$, then$$y(...
Let $$y=f(x)$$ be the solution of the differential equation $$y(x+1)dx-x^2dy=0,y(1)=e$$. Then $$\mathop {\lim }\limits_{x \to {0^ + }} f(x)$$ is equal...
Let $$y=y(t)$$ be a solution of the differential equation $${{dy} \over {dt}} + \alpha y = \gamma {e^{ - \beta t}}$$ where, $$\alpha > 0,\beta > 0$$...
Let $$y = y(x)$$ be the solution curve of the differential equation $${{dy} \over {dx}} = {y \over x}\left( {1 + x{y^2}(1 + {{\log }_e}x)} \right),x >... Let$$y=y(x)$$be the solution of the differential equation$$(x^2-3y^2)dx+3xy~dy=0,y(1)=1$$. Then$$6y^2(e)$$is equal to Let$$y = y(x)$$be the solution of the differential equation$${x^3}dy + (xy - 1)dx = 0,x > 0,y\left( {{1 \over 2}} \right) = 3 - \mathrm{e}$$. Then ... If the solution curve of the differential equation$$\frac{d y}{d x}=\frac{x+y-2}{x-y}$$passes through the points$$(2,1)$$and$$(\mathrm{k}+1,2), \...
Let $$y=y(x)$$ be the solution curve of the differential equation $$\frac{d y}{d x}+\left(\frac{2 x^{2}+11 x+13}{x^{3}+6 x^{2}+11 x+6}\right) y=\frac... Let the solution curve$$y=y(x)$$of the differential equation$$\left(1+\mathrm{e}^{2 x}\right)\left(\frac{\mathrm{d} y}{\mathrm{~d} x}+y\right)=1$$... Let$$y=y(x)$$be the solution curve of the differential equation$$ \frac{d y}{d x}+\frac{1}{x^{2}-1} y=\left(\frac{x-1}{x+1}\right)^{1 / 2}$$,$$x >...
The differential equation of the family of circles passing through the points $$(0,2)$$ and $$(0,-2)$$ is :
Let the solution curve of the differential equation $$x \mathrm{~d} y=\left(\sqrt{x^{2}+y^{2}}+y\right) \mathrm{d} x, x>0$$, intersect the line $$x=1... If$$y=y(x), x \in(0, \pi / 2)$$be the solution curve of the differential equation$$\left(\sin ^{2} 2 x\right) \frac{d y}{d x}+\left(8 \sin ^{2} 2 x...
Let $$y=y_{1}(x)$$ and $$y=y_{2}(x)$$ be two distinct solutions of the differential equation $$\frac{d y}{d x}=x+y$$, with $$y_{1}(0)=0$$ and $$y_{2}(... Let the solution curve$$y=f(x)$$of the differential equation$$ \frac{d y}{d x}+\frac{x y}{x^{2}-1}=\frac{x^{4}+2 x}{\sqrt{1-x^{2}}}$$,$$x\in(-1,1)...
If $${{dy} \over {dx}} + 2y\tan x = \sin x,\,0 ... Let a smooth curve$$y=f(x)$$be such that the slope of the tangent at any point$$(x, y)$$on it is directly proportional to$$\left(\frac{-y}{x}\rig...
The slope of the tangent to a curve $$C: y=y(x)$$ at any point $$(x, y)$$ on it is $$\frac{2 \mathrm{e}^{2 x}-6 \mathrm{e}^{-x}+9}{2+9 \mathrm{e}^{-2 ... The general solution of the differential equation$$\left(x-y^{2}\right) \mathrm{d} x+y\left(5 x+y^{2}\right) \mathrm{d} y=0$$is : Let$${{dy} \over {dx}} = {{ax - by + a} \over {bx + cy + a}},\,a,b,c \in R$$, represents a circle with center ($$\alpha$$,$$\beta$$). Then,$$\alpha...
If y = y(x) is the solution of the differential equation $$\left( {1 + {e^{2x}}} \right){{dy} \over {dx}} + 2\left( {1 + {y^2}} \right){e^x} = 0$$ and...
Let the solution curve of the differential equation $$x{{dy} \over {dx}} - y = \sqrt {{y^2} + 16{x^2}}$$, $$y(1) = 3$$ be $$y = y(x)$$. Then y(2) is ...
Let x = x(y) be the solution of the differential equation $$2y\,{e^{x/{y^2}}}dx + \left( {{y^2} - 4x{e^{x/{y^2}}}} \right)dy = 0$$ such that x(1) = 0....
Let the slope of the tangent to a curve y = f(x) at (x, y) be given by 2 $$\tan x(\cos x - y)$$. If the curve passes through the point $$\left( {{\pi ... Let the solution curve$$y = y(x)$$of the differential equation$$\left[ {{x \over {\sqrt {{x^2} - {y^2}} }} + {e^{{y \over x}}}} \right]x{{dy} \over...
Let y = y(x) be the solution of the differential equation $$x(1 - {x^2}){{dy} \over {dx}} + (3{x^2}y - y - 4{x^3}) = 0$$, $$x > 1$$, with $$y(2) = - ... If the solution curve of the differential equation$$(({\tan ^{ - 1}}y) - x)dy = (1 + {y^2})dx$$passes through the point (1, 0), then the abscissa of... Let$${{dy} \over {dx}} = {{ax - by + a} \over {bx + cy + a}}$$, where a, b, c are constants, represent a circle passing through the point (2, 5). The... If$${{dy} \over {dx}} + {{{2^{x - y}}({2^y} - 1)} \over {{2^x} - 1}} = 0$$, x, y > 0, y(1) = 1, then y(2) is equal to : If$$y = y(x)$$is the solution of the differential equation$$x{{dy} \over {dx}} + 2y = x\,{e^x}$$,$$y(1) = 0$$then the local maximum value of the ... If the solution of the differential equation$${{dy} \over {dx}} + {e^x}\left( {{x^2} - 2} \right)y = \left( {{x^2} - 2x} \right)\left( {{x^2} - 2} \r...
If $$y = y(x)$$ is the solution of the differential equation $$2{x^2}{{dy} \over {dx}} - 2xy + 3{y^2} = 0$$ such that $$y(e) = {e \over 3}$$, then y(1...
Let $$g:(0,\infty ) \to R$$ be a differentiable function such that $$\int {\left( {{{x(\cos x - \sin x)} \over {{e^x} + 1}} + {{g(x)\left( {{e^x} + 1 ... Let$$y = y(x)$$be the solution of the differential equation$$(x + 1)y' - y = {e^{3x}}{(x + 1)^2}$$, with$$y(0) = {1 \over 3}$$. Then, the point$$...
If the solution curve $$y = y(x)$$ of the differential equation $${y^2}dx + ({x^2} - xy + {y^2})dy = 0$$, which passes through the point (1, 1) and in...
If x = x(y) is the solution of the differential equation $$y{{dx} \over {dy}} = 2x + {y^3}(y + 1){e^y},\,x(1) = 0$$; then x(e) is equal to :...
If y = y(x) is the solution curve of the differential equation $${x^2}dy + \left( {y - {1 \over x}} \right)dx = 0$$ ; x > 0 and y(1) = 1, then $$y\... If$${{dy} \over {dx}} = {{{2^x}y + {2^y}{{.2}^x}} \over {{2^x} + {2^{x + y}}{{\log }_e}2}}$$, y(0) = 0, then for y = 1, the value of x lies in the in... If$$y{{dy} \over {dx}} = x\left[ {{{{y^2}} \over {{x^2}}} + {{\phi \left( {{{{y^2}} \over {{x^2}}}} \right)} \over {\phi '\left( {{{{y^2}} \over {{x^...
If $${{dy} \over {dx}} = {{{2^{x + y}} - {2^x}} \over {{2^y}}}$$, y(0) = 1, then y(1) is equal to :
A differential equation representing the family of parabolas with axis parallel to y-axis and whose length of latus rectum is the distance of the poin...
If the solution curve of the differential equation (2x $$-$$ 10y3)dy + ydx = 0, passes through the points (0, 1) and (2, $$\beta$$), then $$\beta$$ is...
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 that y(0) = 7. Then y($$\pi$$) is...
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 point (x, f(x)) is given by f...
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 equal to :...
Let y = y(x) be a solution curve of the differential equation $$(y + 1){\tan ^2}x\,dx + \tan x\,dy + y\,dx = 0$$, $$x \in \left( {0,{\pi \over 2}} \r... Let y = y(x) be the solution of the differential equation (x$$-$$x3)dy = (y + yx2$$-$$3x4)dx, x > 2. If y(3) = 3, then y(4) is equal to :... Let y = y(x) be solution of the differential equation$${\log _{}}\left( {{{dy} \over {dx}}} \right) = 3x + 4y$$, with y(0) = 0.If$$y\left( { - {2 \o...
Let y = y(x) be the solution of the differential equation xdy = (y + x3 cosx)dx with y($$\pi$$) = 0, then $$y\left( {{\pi \over 2}} \right)$$ is equa...
Let y = y(x) be the solution of the differential equation $${{dy} \over {dx}} = 1 + x{e^{y - x}}, - \sqrt 2 < x < \sqrt 2 ,y(0) = 0$$then, the ...
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 $$y\left( {{\pi \over 4}} \ri... Let y = y(x) satisfies the equation$${{dy} \over {dx}} - |A| = 0$$, for all x > 0, where$$A = \left[ {\matrix{ y & {\sin x} & 1 \cr ...
Let y = y(x) be the solution of the differential equation $$x\tan \left( {{y \over x}} \right)dy = \left( {y\tan \left( {{y \over x}} \right) - x} \ri... Let y = y(x) be the solution of the differential equation$${e^x}\sqrt {1 - {y^2}} dx + \left( {{y \over x}} \right)dy = 0$$, y(1) =$$-$$1. Then the ... Let y = y(x) be the solution of the differential equation$${{dy} \over {dx}} = (y + 1)\left( {(y + 1){e^{{x^2}/2}} - x} \right)$$, 0 < x < 2.1,... The differential equation satisfied by the system of parabolas y2 = 4a(x + a) is : If the curve y = y(x) is the solution of the differential equation$$2({x^2} + {x^{5/4}})dy - y(x + {x^{1/4}})dx = {2x^{9/4}}dx$$, x > 0 which pass... Let y = y(x) be the solution of the differential equation$$\cos x(3\sin x + \cos x + 3)dy = (1 + y\sin x(3\sin x + \cos x + 3))dx,0 \le x \le {\pi \...
Which of the following is true for y(x) that satisfies the differential equation $${{dy} \over {dx}}$$ = xy $$-$$ 1 + x $$-$$ y; y(0) = 0 :
If y = y(x) is the solution of the differential equation $${{dy} \over {dx}}$$ + (tan x) y = sin x, $$0 \le x \le {\pi \over 3}$$, with y(0) = 0, the...
Let C1 be the curve obtained by the solution of differential equation $$2xy{{dy} \over {dx}} = {y^2} - {x^2},x > 0$$. Let the curve C2 be the solut...
If y = y(x) is the solution of the differential equation, $${{dy} \over {dx}} + 2y\tan x = \sin x,y\left( {{\pi \over 3}} \right) = 0$$, then the max...
The rate of growth of bacteria in a culture is proportional to the number of bacteria present and the bacteria count is 1000 at initial time t = 0. Th...
If a curve passes through the origin and the slope of the tangent to it at any point (x, y) is $${{{x^2} - 4x + y + 8} \over {x - 2}}$$, then this cur...
If a curve y = f(x) passes through the point (1, 2) and satisfies $$x {{dy} \over {dx}} + y = b{x^4}$$, then for what value of b, $$\int\limits_1^2 {f... The population P = P(t) at time 't' of a certain species follows the differential equation$${{dP} \over {dt}}$$= 0.5P – 450. If P(0) = 850, then the... If$$y = \left( {{2 \over \pi }x - 1} \right) cosec\,x$$is the solution of the differential equation,$${{dy} \over {dx}} + p\left( x \right)y = {2 ...
The general solution of the differential equation $$\sqrt {1 + {x^2} + {y^2} + {x^2}{y^2}}$$ + xy$${{dy} \over {dx}}$$ = 0 is : (where C is a constan...
Let y = y(x) be the solution of the differential equation cosx$${{dy} \over {dx}}$$ + 2ysinx = sin2x, x $$\in$$ $$\left( {0,{\pi \over 2}} \right)... If y = y(x) is the solution of the differential equation$${{5 + {e^x}} \over {2 + y}}.{{dy} \over {dx}} + {e^x} = 0$$satisfying y(0) = 1, then a va... The solution of the differential equation$${{dy} \over {dx}} - {{y + 3x} \over {{{\log }_e}\left( {y + 3x} \right)}} + 3 = 0$$is: (where c is a con... Let y = y(x) be the solution of the differential equation, xy'- y = x2(xcosx + sinx), x > 0. if y ($$\pi $$) =$$\pi $$then$$y''\left( {{\pi \ov...
If x3dy + xy dx = x2dy + 2y dx; y(2) = e and x > 1, then y(4) is equal to :
The solution curve of the differential equation, (1 + e-x)(1 + y2)$${{dy} \over {dx}}$$ = y2, which passes through the point (0, 1), is :...
If a curve y = f(x), passing through the point (1, 2), is the solution of the differential equation, 2x2dy= (2xy + y2)dx, then $$f\left( {{1 \over 2}}... Let y = y(x) be the solution of the differential equation,$${{2 + \sin x} \over {y + 1}}.{{dy} \over {dx}} = - \cos x$$, y > 0,y(0) = 1. If y($$\...
If $${{dy} \over {dx}} = {{xy} \over {{x^2} + {y^2}}}$$; y(1) = 1; then a value of x satisfying y(x) = e is :
The differential equation of the family of curves, x2 = 4b(y + b), b $$\in$$ R, is :
Let y = y(x) be a solution of the differential equation, $$\sqrt {1 - {x^2}} {{dy} \over {dx}} + \sqrt {1 - {y^2}} = 0$$, |x| < 1. If $$y\left( {{... Let y = y(x) be the solution curve of the differential equation,$$\left( {{y^2} - x} \right){{dy} \over {dx}} = 1$$, satisfying y(0) = 1. This curve ... If y = y(x) is the solution of the differential equation,$${e^y}\left( {{{dy} \over {dx}} - 1} \right) = {e^x}$$such that y(0) = 0, then y(1) is equ... The general solution of the differential equation (y2 – x3)dx – xydy = 0 (x$$ \ne $$0) is : (where c is a constant of integration)... Consider the differential equation,$${y^2}dx + \left( {x - {1 \over y}} \right)dy = 0$$, If value of y is 1 when x = 1, then the value of x for which... Let y = y(x) be the solution of the differential equation,$${{dy} \over {dx}} + y\tan x = 2x + {x^2}\tan x$$,$$x \in \left( { - {\pi \over 2},{\pi ...
If y = y(x) is the solution of the differential equation $${{dy} \over {dx}} = \left( {\tan x - y} \right){\sec ^2}x$$, $$x \in \left( { - {\pi \ove... If$$\cos x{{dy} \over {dx}} - y\sin x = 6x$$, (0 < x <$${\pi \over 2}$$) and$$y\left( {{\pi \over 3}} \right)$$= 0 then$$y\left( {{\pi \...
The solution of the differential equation $$x{{dy} \over {dx}} + 2y$$ = x2 (x $$\ne$$ 0) with y(1) = 1, is :
Let y = y(x) be the solution of the differential equation, $${({x^2} + 1)^2}{{dy} \over {dx}} + 2x({x^2} + 1)y = 1$$ such that y(0) = 0. If $$\sqrt ay... If a curve passes through the point (1, –2) and has slope of the tangent at any point (x, y) on it as$${{{x^2} - 2y} \over x}$$, then the curve also ... Let y = y(x) be the solution of the differential equation, x$${{dy} \over {dx}}$$+ y = x loge x, (x > 1). If 2y(2) = loge 4$$-$$1, then y(e) i... The solution of the differential equation,$${{dy} \over {dx}}$$= (x – y)2, when y(1) = 1, is : If y(x) is the solution of the differential equation$${{dy} \over {dx}} + \left( {{{2x + 1} \over x}} \right)y = {e^{ - 2x}},\,\,x > 0,\,$$w... The curve amongst the family of curves represented by the differential equation, (x2 – y2)dx + 2xy dy = 0 which passes through (1, 1) is :... Let f be a differentiable function such that f '(x) = 7 -$${3 \over 4}{{f\left( x \right)} \over x},$$(x > 0) and f(1)$$ \ne $$4. Then$$\matho...
If  $${{dy} \over {dx}} + {3 \over {{{\cos }^2}x}}y = {1 \over {{{\cos }^2}x}},\,\,x \in \left( {{{ - \pi } \over 3},{\pi \over 3}} \right)$$&nb...
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 y = y(x) satiesfies the differ...
If y = y(x) is the solution of the differential equation, x$$dy \over dx$$ + 2y = x2, satisfying y(1) = 1, then y($$1\over2$$) is equal to :...
The differential equation representing the family of ellipse having foci eith on the x-axis or on the $$y$$-axis, center at the origin and passing thr...
Let y = y(x) be the solution of the differential equation $$\sin x{{dy} \over {dx}} + y\cos x = 4x$$, $$x \in \left( {0,\pi } \right)$$. If $$y\left( ... The curve satifying the differeial equation, (x2$$-$$y2) dx + 2xydy = 0 and passing through the point (1, 1) is : Let y = y(x) be the solution of the differential equation$${{dy} \over {dx}} + 2y = f\left( x \right),$$where$$f\left( x \right) = \left\{ {\matrix...
If 2x = y$${^{{1 \over 5}}}$$ + y$${^{ - {1 \over 5}}}$$ and (x2 $$-$$ 1) $${{{d^2}y} \over {d{x^2}}}$$ + $$\lambda$$x $${{dy} \over {dx}}$$ + ky = ...
The curve satisfying the differential equation, ydx $$-$$(x + 3y2)dy = 0 and passing through the point (1, 1), also passes through the point : ...
If $$\left( {2 + \sin x} \right){{dy} \over {dx}} + \left( {y + 1} \right)\cos x = 0$$ and y(0) = 1, then $$y\left( {{\pi \over 2}} \right)$$ is equa...
The solution of the differential equation $${{dy} \over {dx}}\, + \,{y \over 2}\,\sec x = {{\tan x} \over {2y}},\,\,$$ where 0 $$\le$$ x < $${\... If f(x) is a differentiable function in the interval (0,$$\infty $$) such that f (1) = 1 and$$\mathop {\lim }\limits_{t \to x} $$&nb... If a curve$$y=f(x)$$passes through the point$$(1,-1)$$and satisfies the differential equation,$$y(1+xy) dx=xdy$$, then$$f\left( { - {1 \ov...
Let $$y(x)$$ be the solution of the differential equation $$\left( {x\,\log x} \right){{dy} \over {dx}} + y = 2x\,\log x,\left( {x \ge 1} \right).$$ ...
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