JEE Main
Mathematics
Differential Equations
Previous Years Questions

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 $\left(3 y^{2}-5 x^{2}\right) y \mathrm{~d} x+2 x\left(x^{2}-y^{2}\right) \mathrm{d} y=0$ ...
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 $$\frac{\mathrm{d} y}{\mathrm{~d} x}-\frac{3 x^{5} \tan ^{-1}\left(x^{3}\right)}{\left... The number of points on the curve$$y=54 x^{5}-135 x^{4}-70 x^{3}+180 x^{2}+210 x$$at which the normal lines are parallel to$$x+90 y+2=0$$is :... Let$$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...
The minimum value of the twice differentiable function $$f(x)=\int\limits_{0}^{x} \mathrm{e}^{x-\mathrm{t}} f^{\prime}(\mathrm{t}) \mathrm{dt}-\left(x... Let$$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 ...
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 :
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 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...
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) 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...
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 :
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...
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 $$y''\left( {{\pi \ov... Let f be a twice differentiable function on (1, 6). If f(2) = 8, f’(2) = 5, f’(x)$$ \ge $$1 and f''(x)$$ \ge $$4, for all x$$ \in $$(1, 6), then... The solution curve of the differential equation, (1 + e-x)(1 + y2)$${{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}}... 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$$, satisfying y(0) = 1. This curve ... Let xk + yk = ak, (a, k > 0 ) and$${{dy} \over {dx}} + {\left( {{y \over x}} \right)^{{1 \over 3}}} = 0$$, then k is:... If y = y(x) is the solution of the differential equation,$${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}}$$= (x – y)2, when y(1) = 1, is : If xloge(logex)$$-$$x2 + y2 = 4(y > 0), then$${{dy} \over {dx}}$$at x = e is equal to : The curve amongst the family of curves represented by the differential equation, (x2 – y2)dx + 2xy dy = 0 which passes through (1, 1) is... Let f be a differentiable function such that f '(x) = 7 -$${3 \over 4}{{f\left( x \right)} \over x},$$(x > 0) and f(1)$$ \ne $$4. Then$$\matho... A helicopter is flying along the curve given by y – x3/2 = 7, (x $$\ge$$ 0). A soldier positioned at the point $$\left( {{1 \over 2},7} \right)$$ wa... 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 maximum volume (in cu.m) of the right circular cone having slant height 3 m is : 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... If the curves y2 = 6x, 9x2 + by2 = 16 intersect each other at right angles, then the value of b is : Let $$f\left( x \right) = {x^2} + {1 \over {{x^2}}}$$ and $$g\left( x \right) = x - {1 \over x}$$, $$x \in R - \left\{ { - 1,0,1} \right\}$$. If $$h\... Let S = { t$$ \in R:f(x) = \left| {x - \pi } \right|.\left( {{e^{\left| x \right|}} - 1} \right)\sin \left| x \right|$$is not differentiable at ... Let y = y(x) be the solution of the differential equation$$\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}}$$+ ky = ... If y =$${\left[ {x + \sqrt {{x^2} - 1} } \right]^{15}} + {\left[ {x - \sqrt {{x^2} - 1} } \right]^{15}},$$then (x2$$-$$1)$${{{d^2}y} \over {d{x^... The curve satisfying the differential equation, ydx $$-$$(x + 3y2)dy = 0 and passing through the point (1, 1), also passes through the point : ... Twenty meters of wire is available for fencing off a flower-bed in the form of a circular sector. Then the maximum area (in sq. m) of the flower-bed, ... 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 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 $$\left( {x\,\log x} \right){{dy} \over {dx}} + y = 2x\,\log x,\left( {x \ge 1} \right).$$ ... Let the population of rabbits surviving at time $$t$$ be governed by the differential equation $${{dp\left( t \right)} \over {dt}} = {1 \over 2}p\lef... At present, a firm is manufacturing$$2000$$items. It is estimated that the rate of change of production P w.r.t. additional number of workers$$x$...
The population $$p$$ $$(t)$$ at time $$t$$ of a certain mouse species satisfies the differential equation $${{dp\left( t \right)} \over {dt}} = 0.5\,p... Let$$I$$be the purchase value of an equipment and$$V(t)$$be the value after it has been used for$$t$$years. The value$$V(t)$$depreciates at a ... If$${{dy} \over {dx}} = y + 3 > 0\,\,$$and$$y(0)=2,$$then$$y\left( {\ln 2} \right)$$is equal to : Solution of the differential equation$$\cos x\,dy = y\left( {\sin x - y} \right)dx,\,\,0 < x{\pi \over 2}$$is The differential equation which represents the family of curves$$y = {c_1}{e^{{c_2}x}},$$where$${c_1}$$, and$${c_2}$$are arbitrary constants, is... The solution of the differential equation$${{dy} \over {dx}} = {{x + y} \over x}$$satisfying the condition$$y(1)=1$$is The differential equation of all circles passing through the origin and having their centres on the$$x$$-axis is The differential equation whose solution is$$A{x^2} + B{y^2} = 1$$where$$A$$and$$B$$are arbitrary constants is of The differential equation representing the family of curves$${y^2} = 2c\left( {x + \sqrt c } \right),$$where$$c>0,$$is a parameter, is of order... If$$x{{dy} \over {dx}} = y\left( {\log y - \log x + 1} \right),$$then the solution of the equation is The differential equation for the family of circle$${x^2} + {y^2} - 2ay = 0,$$where a is an arbitrary constant is Solution of the differential equation$$ydx + \left( {x + {x^2}y} \right)dy = 0$$is The solution of the differential equation$$\left( {1 + {y^2}} \right) + \left( {x - {e^{{{\tan }^{ - 1}}y}}} \right){{dy} \over {dx}} = 0,$$is ... The degree and order of the differential equation of the family of all parabolas whose axis is$$x$$-axis, are respectively. The solution of the equation$$\,{{{d^2}y} \over {d{x^2}}} = {e^{ - 2x}}$$The order and degree of the differential equation$$\,{\left( {1 + 3{{dy} \over {dx}}} \right)^{2/3}} = 4{{{d^3}y} \over {d{x^3}}}$$are ## Numerical Let$$y=y(x)$$be the solution curve of the differential equation$$\sin \left( {2{x^2}} \right){\log _e}\left( {\tan {x^2}} \right)dy + \left( {4xy -...
Suppose $$y=y(x)$$ be the solution curve to the differential equation $$\frac{d y}{d x}-y=2-e^{-x}$$ such that $$\lim\limits_{x \rightarrow \infty} y(... Let$$y=y(x)$$be the solution of the differential equation$$\frac{d y}{d x}=\frac{4 y^{3}+2 y x^{2}}{3 x y^{2}+x^{3}}, y(1)=1$$. If for some$$n \in...
Let $$f$$ be a twice differentiable function on $$\mathbb{R}$$. If $$f^{\prime}(0)=4$$ and $$f(x) + \int\limits_0^x {(x - t)f'(t)dt = \left( {{e^{2x}}... Let y = y(x), x > 1, be the solution of the differential equation$$(x - 1){{dy} \over {dx}} + 2xy = {1 \over {x - 1}}$$, with$$y(2) = {{1 + {e^4}} \...
Let y = y(x) be the solution of the differential equation $${{dy} \over {dx}} + {{\sqrt 2 y} \over {2{{\cos }^4}x - {{\cos }^2}x}} = x{e^{{{\tan }^{ -... Let$$y = y(x)$$be the solution of the differential equation$$(1 - {x^2})dy = \left( {xy + ({x^3} + 2)\sqrt {1 - {x^2}} } \right)dx, - 1 $$-$$1 is e...
Let the solution curve y = y(x) of the differential equation $$(4 + {x^2})dy - 2x({x^2} + 3y + 4)dx = 0$$ pass through the origin. Then y(2) is equal ...
Let $$S = (0,2\pi ) - \left\{ {{\pi \over 2},{{3\pi } \over 4},{{3\pi } \over 2},{{7\pi } \over 4}} \right\}$$. Let $$y = y(x)$$, x $$\in$$ S, be the...
If $${y^{1/4}} + {y^{ - 1/4}} = 2x$$, and $$({x^2} - 1){{{d^2}y} \over {d{x^2}}} + \alpha x{{dy} \over {dx}} + \beta y = 0$$, then | $$\alpha$$ $$-$$ ...
Let y = y(x) be the solution of the differential equation dy = e$$\alpha$$x + y dx; $$\alpha$$ $$\in$$ N. If y(loge2) = loge2 and y(0) = loge$$\left( ... Let$$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} + 2t + 4F'(t))dt} $$. If$$F'...
If $$y = y(x),y \in \left[ {0,{\pi \over 2}} \right)$$ is the solution of the differential equation $$\sec y{{dy} \over {dx}} - \sin (x + y) - \sin (... Let y = y(x) be solution of the following differential equation$${e^y}{{dy} \over {dx}} - 2{e^y}\sin x + \sin x{\cos ^2}x = 0,y\left( {{\pi \over 2}...
Let y = y(x) be the solution of the differential equation $$\left( {(x + 2){e^{\left( {{{y + 1} \over {x + 2}}} \right)}} + (y + 1)} \right)dx = (x + ... Let a curve y = y(x) be given by the solution of the differential equation$$\cos \left( {{1 \over 2}{{\cos }^{ - 1}}({e^{ - x}})} \right)dx = \sqrt {...
Let y = y(x) be the solution of the differential equation xdy $$-$$ ydx = $$\sqrt {({x^2} - {y^2})} dx$$, x $$\ge$$ 1, with y(1) = 0. If the area bo...
The difference between degree and order of a differential equation that represents the family of curves given by $${y^2} = a\left( {x + {{\sqrt a } \o... If y = y(x) is the solution of the equation$${e^{\sin y}}\cos y{{dy} \over {dx}} + {e^{\sin y}}\cos x = \cos x$$, y(0) = 0; then$$1 + y\left( {{\pi ...
If the curve, y = y(x) represented by the solution of the differential equation (2xy2 $$-$$ y)dx + xdy = 0, passes through the intersection of the lin...
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, then y(3) is equal to _______....
EXAM MAP
Joint Entrance Examination