JEE MAIN · MATHEMATICS · DIFFERENTIAL EQUATIONS
JEE Main Differential Equations Questions & Solutions
172 solved questions on Differential Equations, ranging from easy to JEE-Advanced-flavour hard. Click any to see the full solution.
172 solved questions on Differential Equations, ranging from easy to JEE-Advanced-flavour hard. Click any to see the full solution.
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 equal to :
View solution →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 $x\left(e^{4}-2\right)=1$. Then…
View solution →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 interval :
View solution →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 $y(2) = - 2$. Then y(3) is equal to :
View solution →Let $y=y(x)$ be the solution curve of the differential equation $$\sin \left( {2{x^2}} \right){\log _e}\left( {\tan {x^2}} \right)dy + \left( {4xy - 4\sqrt 2 x\sin \left( {{x^2} -…
View solution →The general solution of the differential equation $\left(x-y^{2}\right) \mathrm{d} x+y\left(5 x+y^{2}\right) \mathrm{d} y=0$ is :
View solution →Suppose $y=y(x)$ be the solution curve to the differential equation $\frac{d y}{d x}-y=2-e^{-x}$ such that $\lim\limits_{x \rightarrow \infty} y(x)$ is finite. If $a$ and $b$ are…
View solution →Let $\alpha x=\exp \left(x^{\beta} y^{\gamma}\right)$ be the solution of the differential equation $$2 x^{2} y \mathrm{~d} y-\left(1-x y^{2}\right) \mathrm{d} x=0, x…
View solution →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 passes through the point $\left( {1,1…
View solution →Let y = y(x) be the solution of the differential equation dy = e$\alpha$x + y dx; $\alpha$ $\in$ N. If y(loge2) = loge2 and y(0) = loge$\left( {{1 \over 2}} \right)$, then the…
View solution →Let y = y(x) be the solution of the differential equation $${{dy} \over {dx}} + {{\sqrt 2 y} \over {2{{\cos }^4}x - {{\cos }^2}x}} = x{e^{{{\tan }^{ - 1}}(\sqrt 2 \cot 2x)}},\,0 2…
View solution →The temperature $T(t)$ of a body at time $t=0$ is $160^{\circ} \mathrm{F}$ and it decreases continuously as per the differential equation $\frac{d T}{d t}=-K(T-80)$, where $K$ is…
View solution →Let $y=y(x)$ be the solution of the differential equation $$\frac{\mathrm{d} y}{\mathrm{~d} x}+\frac{2 x}{\left(1+x^2\right)^2} y=x \mathrm{e}^{\frac{1}{\left(1+x^2\right)}} ;…
View solution →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 $y\left( {{\pi \over 4}} \right) = 0$. Then, the value of…
View solution →If the solution curve, of the differential equation $\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{x+y-2}{x-y}$ passing through the point $(2,1)$ is $$\tan…
View solution →Let the slope of the tangent to a curve y = f(x) at (x, y) be given by 2 $\tan x(\cos x - y)$. If the curve passes through the point $\left( {{\pi \over 4},0} \right)$, then the…
View solution →If the solution curve of the differential equation (2x $-$ 10y3)dy + ydx = 0, passes through the points (0, 1) and (2, $\beta$), then $\beta$ is a root of the equation :
View solution →Let y = y(x) be the solution curve of the differential equation, $\left( {{y^2} - x} \right){{dy} \over {dx}} = 1$, satisfying y(0) = 1. This curve intersects the x-axis at a…
View solution →Let $y=y(t)$ be a solution of the differential equation ${{dy} \over {dt}} + \alpha y = \gamma {e^{ - \beta t}}$ where, $\alpha 0,\beta 0$ and $\gamma 0$. Then $\mathop {\lim…
View solution →Let a differentiable function $f$ satisfy $f(x)+\int_\limits{3}^{x} \frac{f(t)}{t} d t=\sqrt{x+1}, x \geq 3$. Then $12 f(8)$ is equal to :
View solution →If ${{dy} \over {dx}} = {{{2^{x + y}} - {2^x}} \over {{2^y}}}$, y(0) = 1, then y(1) is equal to :
View solution →Let $y=y(x)$ be the solution of the differential equation $\sec x \mathrm{~d} y+\{2(1-x) \tan x+x(2-x)\} \mathrm{d} x=0$ such that $y(0)=2$. Then $y(2)$ is equal to:
View solution →Let $$S = (0,2\pi ) - \left\{ {{\pi \over 2},{{3\pi } \over 4},{{3\pi } \over 2},{{7\pi } \over 4}} \right\}$$. Let $y = y(x)$, x $\in$ S, be the solution curve of the…
View solution →If $y=y(x)$ is the solution curve of the differential equation $$\left(x^2-4\right) \mathrm{d} y-\left(y^2-3 y\right) \mathrm{d} x=0, x2, y(4)=\frac{3}{2}$$ and the slope of the…
View solution →Let $y=y(x)$ be the solution of the differential equation $$\frac{d y}{d x}=\frac{(\tan x)+y}{\sin x(\sec x-\sin x \tan x)}, x \in\left(0, \frac{\pi}{2}\right)$$ satisfying the…
View solution →Let $y=y(x)$ be the solution of the differential equation $x{\log _e}x{{dy} \over {dx}} + y = {x^2}{\log _e}x,(x 1)$. If $y(2) = 2$, then $y(e)$ is equal to
View solution →The solution curve of the differential equation $y \frac{d x}{d y}=x\left(\log _e x-\log _e y+1\right), x0, y0$ passing through the point $(e, 1)$ is
View solution →Let $y=y(x)$ be the solution of the differential equation $\frac{d y}{d x}=\frac{4 y^{3}+2 y x^{2}}{3 x y^{2}+x^{3}}, y(1)=1$. If for some $n \in \mathbb{N}, y(2) \in[n-1, n)$,…
View solution →Let $y=y(x)$ be the solution of the differential equation $(x^2-3y^2)dx+3xy~dy=0,y(1)=1$. Then $6y^2(e)$ is equal to
View solution →If a curve y = f(x) passes through the point (1, 2) and satisfies $x {{dy} \over {dx}} + y = b{x^4}$, then for what value of b, $\int\limits_1^2 {f(x)dx = {{62} \over 5}}$?
View solution →A function $y=f(x)$ satisfies $f(x) \sin 2 x+\sin x-\left(1+\cos ^2 x\right) f^{\prime}(x)=0$ with condition $f(0)=0$. Then, $f\left(\frac{\pi}{2}\right)$ is equal to
View solution →If y = y(x) is the solution of the equation ${e^{\sin y}}\cos y{{dy} \over {dx}} + {e^{\sin y}}\cos x = \cos x$, y(0) = 0; then $$1 + y\left( {{\pi \over 6}} \right) + {{\sqrt 3 }…
View solution →Let $y=y(x)$ be the solution of the differential equation $(1+y^2) e^{\tan x} d x+\cos ^2 x(1+e^{2 \tan x}) d y=0, y(0)=1$. Then $y\left(\frac{\pi}{4}\right)$ is equal to
View solution →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 value of (y(3))2 is equal to :
View solution →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 lines, 2x $-$ 3y = 1 and 3x + 2y =…
View solution →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 $\left(\frac{-y}{x}\right)$. If the curve passes through…
View solution →Let $y=f(x)$ be the solution of the differential equation $\frac{\mathrm{d} y}{\mathrm{~d} x}+\frac{x y}{x^2-1}=\frac{x^6+4 x}{\sqrt{1-x^2}},-1
View solution →The differential equation of the family of circles passing through the points $(0,2)$ and $(0,-2)$ is :
View solution →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 bounded by the line x = 1, x = e$\pi$,…
View solution →Let the solution curve of the differential equation $x \mathrm{~d} y=\left(\sqrt{x^{2}+y^{2}}+y\right) \mathrm{d} x, x0$, intersect the line $x=1$ at $y=0$ and the line $x=2$ at…
View solution →Let $g:(0,\infty ) \to R$ be a differentiable function such that $$\int {\left( {{{x(\cos x - \sin x)} \over {{e^x} + 1}} + {{g(x)\left( {{e^x} + 1 - x{e^x}} \right)} \over…
View solution →Let y = y(x) be the solution of the differential equation ${{dy} \over {dx}} = (y + 1)\left( {(y + 1){e^{{x^2}/2}} - x} \right)$, 0 < x < 2.1, with y(2) = 0. Then the value of…
View solution →Let $y=f(x)$ be the solution of the differential equation $y(x+1)dx-x^2dy=0,y(1)=e$. Then $\mathop {\lim }\limits_{x \to {0^ + }} f(x)$ is equal to
View solution →If the solution curve $y = y(x)$ of the differential equation ${y^2}dx + ({x^2} - xy + {y^2})dy = 0$, which passes through the point (1, 1) and intersects the line $y = \sqrt 3 x$…
View solution →If $y = y(x),y \in \left[ {0,{\pi \over 2}} \right)$ is the solution of the differential equation $\sec y{{dy} \over {dx}} - \sin (x + y) - \sin (x - y) = 0$, with y(0) = 0, then…
View solution →Let y = y(x) be a solution of the differential equation, $\sqrt {1 - {x^2}} {{dy} \over {dx}} + \sqrt {1 - {y^2}} = 0$, |x| < 1. If $y\left( {{1 \over 2}} \right) = {{\sqrt 3 }…
View solution →The rate of growth of bacteria in a culture is proportional to the number of bacteria present and the bacteria count is 1000 at initial time t = 0. The number of bacteria is…
View solution →If ${y^{1/4}} + {y^{ - 1/4}} = 2x$, and $({x^2} - 1){{{d^2}y} \over {d{x^2}}} + \alpha x{{dy} \over {dx}} + \beta y = 0$, then | $\alpha$ $-$ $\beta$ | is equal to __________.
View solution →Let the solution curve $y = y(x)$ of the differential equation $$\left[ {{x \over {\sqrt {{x^2} - {y^2}} }} + {e^{{y \over x}}}} \right]x{{dy} \over {dx}} = x + \left[ {{x \over…
View solution →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 $y_{1}(0)=0$ and $y_{2}(0)=1$ respectively.…
View solution →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 time at which population…
View solution →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 $x = - {4 \over 3}$ for the curve $y…
View solution →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 the point on the curve whose…
View solution →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 equal to:
View solution →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 constant of integration)
View solution →Which of the following is true for y(x) that satisfies the differential equation ${{dy} \over {dx}}$ = xy $-$ 1 + x $-$ y; y(0) = 0 :
View solution →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 equal to :
View solution →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 :
View solution →Let a curve y = f(x) pass through the point (2, (loge2)2) and have slope ${{2y} \over {x{{\log }_e}x}}$ for all positive real value of x. Then the value of f(e) is equal to…
View solution →Let y = y(x) be solution of the following differential equation $${e^y}{{dy} \over {dx}} - 2{e^y}\sin x + \sin x{\cos ^2}x = 0,y\left( {{\pi \over 2}} \right) = 0$$ If $y(0) =…
View solution →Let $y=y(x)$ be the solution curve of the differential equation $$ \frac{d y}{d x}+\left(\frac{2 x^{2}+11 x+13}{x^{3}+6 x^{2}+11 x+6}\right) y=\frac{(x+3)}{x+1}, x-1$$, which…
View solution →The solution of the differential equation $(x^2+y^2) \mathrm{d} x-5 x y \mathrm{~d} y=0, y(1)=0$, is :
View solution →Let a curve $y=y(x)$ pass through the point $(3,3)$ and the area of the region under this curve, above the $x$-axis and between the abscissae 3 and $x(3)$ be…
View solution →Let $y=y(x)$ be the solution of the differential equation $$\frac{d y}{d x}+\frac{5}{x\left(x^{5}+1\right)} y=\frac{\left(x^{5}+1\right)^{2}}{x^{7}}, x 0$$. If $y(1)=2$, then…
View solution →The solution curve, of the differential equation $2 y \frac{\mathrm{d} y}{\mathrm{~d} x}+3=5 \frac{\mathrm{d} y}{\mathrm{~d} x}$, passing through the point $(0,1)$ is a conic,…
View solution →Let $y=y(x)$ be the solution of the differential equation $(x^2+4)^2 d y+(2 x^3 y+8 x y-2) d x=0$. If $y(0)=0$, then $y(2)$ is equal to
View solution →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 function $z(x) = {x^2}y(x) -…
View solution →Let a curve y = y(x) be given by the solution of the differential equation $$\cos \left( {{1 \over 2}{{\cos }^{ - 1}}({e^{ - x}})} \right)dx = \sqrt {{e^{2x}} - 1} dy$$. If it…
View solution →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 $y(2 \sqrt{2})=\beta$, then
View solution →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 $y_{2}(0)=1$ respectively. Then, the number…
View solution →If $y = \left( {{2 \over \pi }x - 1} \right) cosec\,x$ is the solution of the differential equation, ${{dy} \over {dx}} + p\left( x \right)y = {2 \over \pi } cosec\,x$, $0 < x <…
View solution →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 value of y(loge13) is :
View solution →If a curve y = f(x), passing through the point (1, 2), is the solution of the differential equation, 2x2dy= (2xy + y2)dx, then $f\left( {{1 \over 2}} \right)$ is equal to :
View solution →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 equal to:
View solution →Let y = y(x) be the solution of the differential equation $$\left( {(x + 2){e^{\left( {{{y + 1} \over {x + 2}}} \right)}} + (y + 1)} \right)dx = (x + 2)dy$$, y(1) = 1. If the…
View solution →Let $x=x(y)$ be the solution of the differential equation $y^2 \mathrm{~d} x+\left(x-\frac{1}{y}\right) \mathrm{d} y=0$. If $x(1)=1$, then $x\left(\frac{1}{2}\right)$ is :
View solution →Let the solution curve $y=f(x)$ of the differential equation $\frac{d y}{d x}+\frac{x y}{x^{2}-1}=\frac{x^{4}+2 x}{\sqrt{1-x^{2}}}$, $x\in(-1,1)$ pass through the origin. Then…
View solution →If ${{dy} \over {dx}} = {{xy} \over {{x^2} + {y^2}}}$; y(1) = 1; then a value of x satisfying y(x) = e is :
View solution →Let $y=y(x)$ be the solution curve of the differential equation $\sec y \frac{\mathrm{d} y}{\mathrm{~d} x}+2 x \sin y=x^3 \cos y, y(1)=0$. Then $y(\sqrt{3})$ is equal to:
View solution →If x3dy + xy dx = x2dy + 2y dx; y(2) = e and x > 1, then y(4) is equal to :
View solution →Suppose the solution of the differential equation $$\frac{d y}{d x}=\frac{(2+\alpha) x-\beta y+2}{\beta x-2 \alpha y-(\beta \gamma-4 \alpha)}$$ represents a circle passing through…
View solution →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 :
View solution →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 \over 2}} \right) + y\left( {{\pi \over 2}}…
View solution →Let $y=y(x)$ be the solution of the differential equation $\left(1+x^2\right) \frac{d y}{d x}+y=e^{\tan ^{-1} x}$, $y(1)=0$. Then $y(0)$ is
View solution →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 $4 \pi$. Let $P$ and $Q$ be the points of…
View solution →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 :
View solution →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 $(\mathrm{k}+1,2), \mathrm{k}0$, then
View solution →If $\sin \left(\frac{y}{x}\right)=\log _e|x|+\frac{\alpha}{2}$ is the solution of the differential equation $$x \cos \left(\frac{y}{x}\right) \frac{d y}{d x}=y \cos…
View solution →Let $f$ be a differentiable function such that ${x^2}f(x) - x = 4\int\limits_0^x {tf(t)dt}$, $f(1) = {2 \over 3}$. Then $18f(3)$ is equal to :
View solution →Let y = y(x) be a solution curve of the differential equation $(y + 1){\tan ^2}x\,dx + \tan x\,dy + y\,dx = 0$, $x \in \left( {0,{\pi \over 2}} \right)$. If $\mathop {\lim…
View solution →Let y = y(x) satisfies the equation ${{dy} \over {dx}} - |A| = 0$, for all x > 0, where $$A = \left[ {\matrix{ y & {\sin x} & 1 \cr 0 & { - 1} & 1 \cr 2 & 0 & {{1 \over x}} \cr }…
View solution →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 :
View solution →The solution curve of the differential equation, (1 + e-x)(1 + y2)${{dy} \over {dx}}$ = y2, which passes through the point (0, 1), is :
View solution →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, then $y\left( {{\pi \over 4}}…
View solution →Let $y=y(x)$ be the solution of the differential equation $$\sec ^2 x d x+\left(e^{2 y} \tan ^2 x+\tan x\right) d y=0,0If $y(\pi / 6)=\alpha$, then $e^{8 \alpha}$ is equal to…
View solution →If the solution $y(x)$ of the given differential equation $\left(e^y+1\right) \cos x \mathrm{~d} x+\mathrm{e}^y \sin x \mathrm{~d} y=0$ passes through the point…
View solution →Let ${{dy} \over {dx}} = {{ax - by + a} \over {bx + cy + a}}$, where a, b, c are constants, represent a circle passing through the point (2, 5). Then the shortest distance of the…
View solution →Let $\alpha|x|=|y| \mathrm{e}^{x y-\beta}, \alpha, \beta \in \mathbf{N}$ be the solution of the differential equation $x \mathrm{~d} y-y \mathrm{~d} x+x y(x \mathrm{~d} y+y…
View solution →If x = x(y) is the solution of the differential equation $y{{dx} \over {dy}} = 2x + {y^3}(y + 1){e^y},\,x(1) = 0$; then x(e) is equal to :
View solution →Let $y=y(x)$ be the solution of the differential equation $$\left(1-x^2\right) \mathrm{d} y=\left[x y+\left(x^3+2\right) \sqrt{3\left(1-x^2\right)}\right] \mathrm{d} x, -1
View solution →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(x) + xf'(x) = x2. Then :
View solution →Let $y=y(x)$ be a solution of the differential equation $(x \cos x) d y+(x y \sin x+y \cos x-1) d x=0,0
View solution →Let $f(x)$ be a positive function such that the area bounded by $y=f(x), y=0$ from $x=0$ to $x=a0$ is $e^{-a}+4 a^2+a-1$. Then the differential equation, whose general solution is…
View solution →If the solution curve $y=y(x)$ of the differential equation $\left(1+y^2\right)\left(1+\log _{\mathrm{e}} x\right) d x+x d y=0, x 0$ passes through the point $(1,1)$ and…
View solution →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) is equal to :
View solution →If $$y{{dy} \over {dx}} = x\left[ {{{{y^2}} \over {{x^2}}} + {{\phi \left( {{{{y^2}} \over {{x^2}}}} \right)} \over {\phi '\left( {{{{y^2}} \over {{x^2}}}} \right)}}} \right]$$, x…
View solution →If the solution of the differential equation $${{dy} \over {dx}} + {e^x}\left( {{x^2} - 2} \right)y = \left( {{x^2} - 2x} \right)\left( {{x^2} - 2} \right){e^{2x}}$$ satisfies…
View solution →If ${{dy} \over {dx}} + 2y\tan x = \sin x,\,0
View solution →Let the solution curve $y=y(x)$ of the differential equation $$\left(1+\mathrm{e}^{2 x}\right)\left(\frac{\mathrm{d} y}{\mathrm{~d} x}+y\right)=1$$ pass through the point…
View solution →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 y (1) is equal to
View solution →If the solution curve of the differential equation $\left(y-2 \log _{e} x\right) d x+\left(x \log _{e} x^{2}\right) d y=0, x 1$ passes through the points $\left(e,…
View solution →If the solution $y=y(x)$ of the differential equation $(x^4+2 x^3+3 x^2+2 x+2) \mathrm{d} y-(2 x^2+2 x+3) \mathrm{d} x=0$ satisfies $y(-1)=-\frac{\pi}{4}$, then $y(0)$ is equal to…
View solution →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 y (0) = 0, then $$6\left(…
View solution →Let $y = y(x)$ be the solution curve of the differential equation $${{dy} \over {dx}} = {y \over x}\left( {1 + x{y^2}(1 + {{\log }_e}x)} \right),x 0,y(1) = 3$$. Then ${{{y^2}(x)}…
View solution →A differential equation representing the family of parabolas with axis parallel to y-axis and whose length of latus rectum is the distance of the point (2, $-$3) from the line 3x…
View solution →Let $y=y(x)$ be the solution curve of the differential equation $\frac{d y}{d x}+\frac{1}{x^{2}-1} y=\left(\frac{x-1}{x+1}\right)^{1 / 2}$, $x 1$ passing through the point…
View solution →Let y = y(x) be the solution of the differential equation $$x\tan \left( {{y \over x}} \right)dy = \left( {y\tan \left( {{y \over x}} \right) - x} \right)dx$$, $- 1 \le x \le 1$,…
View solution →Let $f(x)$ be a real differentiable function such that $f(0)=1$ and $f(x+y)=f(x) f^{\prime}(y)+f^{\prime}(x) f(y)$ for all $x, y \in \mathbf{R}$. Then $\sum_\limits{n=1}^{100}…
View solution →The differential equation of the family of curves, x2 = 4b(y + b), b $\in$ R, is :
View solution →If for x $\ge$ 0, y = y(x) is the solution of the differential equation (x + 1)dy = ((x + 1)2 + y – 3)dx, y(2) = 0, then y(3) is equal to _______.
View solution →If $y=y(x)$ is the solution curve of the differential equation $\frac{d y}{d x}+y \tan x=x \sec x, 0 \leq x \leq \frac{\pi}{3}, y(0)=1$, then $y\left(\frac{\pi}{6}\right)$ is…
View solution →Let y = y(x) be the solution of the differential equation, ${{2 + \sin x} \over {y + 1}}.{{dy} \over {dx}} = - \cos x$, y > 0,y(0) = 1. If y($\pi$) = a and ${{dy} \over {dx}}$ at…
View solution →The difference between degree and order of a differential equation that represents the family of curves given by ${y^2} = a\left( {x + {{\sqrt a } \over 2}} \right)$, a > 0 is…
View solution →The differential equation satisfied by the system of parabolas y2 = 4a(x + a) is :
View solution →Let the solution curve y = y(x) of the differential equation $(4 + {x^2})dy - 2x({x^2} + 3y + 4)dx = 0$ pass through the origin. Then y(2) is equal to _____________.
View solution →If y = y(x) is the solution curve of the differential equation ${x^2}dy + \left( {y - {1 \over x}} \right)dx = 0$ ; x > 0 and y(1) = 1, then $y\left( {{1 \over 2}} \right)$ is…
View solution →Let x = x(y) be the solution of the differential equation $2y\,{e^{x/{y^2}}}dx + \left( {{y^2} - 4x{e^{x/{y^2}}}} \right)dy = 0$ such that x(1) = 0. Then, x(e) is equal to :
View solution →Let y = y(x), x 1, be the solution of the differential equation $(x - 1){{dy} \over {dx}} + 2xy = {1 \over {x - 1}}$, with $y(2) = {{1 + {e^4}} \over {2{e^4}}}$. If $y(3) =…
View solution →Let y = y(x) be the solution of the differential equation $$\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 \over 2},y(0) = 0$$. Then,…
View solution →For a differentiable function $f: \mathbb{R} \rightarrow \mathbb{R}$, suppose $f^{\prime}(x)=3 f(x)+\alpha$, where $\alpha \in \mathbb{R}, f(0)=1$ and $\lim _\limits{x…
View solution →Let y = y(x) be solution of the differential equation ${\log _{}}\left( {{{dy} \over {dx}}} \right) = 3x + 4y$, with y(0) = 0.If $y\left( { - {2 \over 3}{{\log }_e}2} \right) =…
View solution →If $y=y(x), x \in(0, \pi / 2)$ be the solution curve of the differential equation $$\left(\sin ^{2} 2 x\right) \frac{d y}{d x}+\left(8 \sin ^{2} 2 x+2 \sin 4 x\right) y=2…
View solution →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$\left( {{\pi \over 3}}…
View solution →If $y=y(x)$ is the solution of the differential equation $$\frac{d y}{d x}+\frac{4 x}{\left(x^{2}-1\right)} y=\frac{x+2}{\left(x^{2}-1\right)^{\frac{5}{2}}}, x 1$$ such that…
View solution →If $y=y(x)$ is the solution of the differential equation $\frac{\mathrm{d} y}{\mathrm{~d} x}+2 y=\sin (2 x), y(0)=\frac{3}{4}$, then $y\left(\frac{\pi}{8}\right)$ is equal to :
View solution →The slope of the tangent to a curve $C: y=y(x)$ at any point $(x, y)$ on it is $\frac{2 \mathrm{e}^{2 x}-6 \mathrm{e}^{-x}+9}{2+9 \mathrm{e}^{-2 x}}$. If $C$ passes through the…
View solution →Let $y=y(x)$ be the solution of the differential equation $\left(2 x \log _e x\right) \frac{d y}{d x}+2 y=\frac{3}{x} \log _e x, x0$ and $y\left(e^{-1}\right)=0$. Then, $y(e)$ is…
View solution →Let the solution curve $y=y(x)$ of the differential equation $$ \frac{\mathrm{d} y}{\mathrm{~d} x}-\frac{3 x^{5} \tan ^{-1}\left(x^{3}\right)}{\left(1+x^{6}\right)^{3 / 2}} y=2 x…
View solution →Let the solution $y=y(x)$ of the differential equation $\frac{\mathrm{d} y}{\mathrm{~d} x}-y=1+4 \sin x$ satisfy $y(\pi)=1$. Then $y\left(\frac{\pi}{2}\right)+10$ is equal to…
View solution →If the solution curve $f(x, y)=0$ of the differential equation $\left(1+\log _{e} x\right) \frac{d x}{d y}-x \log _{e} x=e^{y}, x 0$, passes through the points $(1,0)$ and…
View solution →Let the solution curve $x=x(y), 0
View solution →If y = y(x) is the solution of the differential equation, ${{dy} \over {dx}} + 2y\tan x = \sin x,y\left( {{\pi \over 3}} \right) = 0$, then the maximum value of the function y(x)…
View solution →Let $y = y(x)$ be the solution of the differential equation $$(1 - {x^2})dy = \left( {xy + ({x^3} + 2)\sqrt {1 - {x^2}} } \right)dx, - 1 $-$1 is equal to _____________.
View solution →Let $y=y(x)$ be the solution of the differential equation $(x+y+2)^2 d x=d y, y(0)=-2$. Let the maximum and minimum values of the function $y=y(x)$ in $\left[0,…
View solution →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 curve $y=y(x)$ at $y=2$ and the line…
View solution →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 constant of integration)
View solution →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 curve also passes through the…
View solution →Let ${{dy} \over {dx}} = {{ax - by + a} \over {bx + cy + a}},\,a,b,c \in R$, represents a circle with center ($\alpha$, $\beta$). Then, $\alpha$ + 2$\beta$ is equal to :
View solution →If the solution of the differential equation $(2 x+3 y-2) \mathrm{d} x+(4 x+6 y-7) \mathrm{d} y=0, y(0)=3$, is $\alpha x+\beta y+3 \log _e|2 x+3 y-\gamma|=6$, then $\alpha+2…
View solution →If $x=x(t)$ is the solution of the differential equation $(t+1) \mathrm{d} x=\left(2 x+(t+1)^4\right) \mathrm{dt}, x(0)=2$, then, $x(1)$ equals _________.
View solution →If $\frac{\mathrm{d} x}{\mathrm{~d} y}=\frac{1+x-y^2}{y}, x(1)=1$, then $5 x(2)$ is equal to __________.
View solution →Let $f$ be a differentiable function such that $2(x+2)^2 f(x)-3(x+2)^2=10 \int_0^x(t+2) f(t) d t, x \geq 0$. Then $f(2)$ is equal to ________ .
View solution →Let $y=y(x)$ be the solution of the differential equation $2 \cos x \frac{\mathrm{~d} y}{\mathrm{~d} x}=\sin 2 x-4 y \sin x, x \in\left(0, \frac{\pi}{2}\right)$. If…
View solution →If $y=y(x)$ is the solution of the differential equation, $\sqrt{4-x^2} \frac{\mathrm{~d} y}{\mathrm{~d} x}=\left(\left(\sin ^{-1}\left(\frac{x}{2}\right)\right)^2-y\right) \sin…
View solution →Let $y=y(x)$ be the solution of the differential equation $\frac{\mathrm{d} y}{\mathrm{~d} x}+2 y \sec ^2 x=2 \sec ^2 x+3 \tan x \cdot \sec ^2 x$ such that $y(0)=\frac{5}{4}$.…
View solution →Let $y=y(x)$ be the solution of the differential equation $\left(3 y^{2}-5 x^{2}\right) y \mathrm{~d} x+2 x\left(x^{2}-y^{2}\right) \mathrm{d} y=0$ such that $y(1)=1$. Then…
View solution →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{\mathrm{d}…
View solution →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,…
View solution →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\limits_{x \rightarrow-\infty}…
View solution →If $x=f(y)$ is the solution of the differential equation $\left(1+y^2\right)+\left(x-2 \mathrm{e}^{\tan ^{-1} y}\right) \frac{\mathrm{d} y}{\mathrm{~d} x}=0, y…
View solution →Let a curve $y=f(x)$ pass through the points $(0,5)$ and $\left(\log _e 2, k\right)$. If the curve satisfies the differential equation $2(3+y) e^{2 x} d x-\left(7+e^{2 x}\right) d…
View solution →Let $x=x(y)$ be the solution of the differential equation $y=\left(x-y \frac{\mathrm{~d} x}{\mathrm{~d} y}\right) \sin \left(\frac{x}{y}\right), y0$ and $x(1)=\frac{\pi}{2}$. Then…
View solution →Let $\mathrm{y}=\mathrm{y}(\mathrm{x})$ be the solution of the differential equation $\left(x y-5 x^2 \sqrt{1+x^2}\right) d x+\left(1+x^2\right) d y=0, y(0)=0$. Then $y(\sqrt{3})$…
View solution →Let for some function $\mathrm{y}=f(x), \int_0^x t f(t) d t=x^2 f(x), x0$ and $f(2)=3$. Then $f(6)$ is equal to
View solution →Let y = y(x) be the solution of the differential equation : $\cos x\left(\log _e(\cos x)\right)^2 d y+\left(\sin x-3 y \sin x \log _e(\cos x)\right) d x=0$, x ∈ (0,…
View solution →If for the solution curve $y=f(x)$ of the differential equation $\frac{d y}{d x}+(\tan x) y=\frac{2+\sec x}{(1+2 \sec x)^2}$, $x \in\left(\frac{-\pi}{2}, \frac{\pi}{2}\right),…
View solution →Let $g$ be a differentiable function such that $\int_0^x g(t) d t=x-\int_0^x \operatorname{tg}(t) d t, x \geq 0$ and let $y=y(x)$ satisfy the differential equation $\frac{d y}{d…
View solution →Let $y=y(x)$ be the solution of the differential equation $\frac{d y}{d x}+3\left(\tan ^2 x\right) y+3 y=\sec ^2 x, y(0)=\frac{1}{3}+e^3$. Then $y\left(\frac{\pi}{4}\right)$ is…
View solution →If a curve $y=y(x)$ passes through the point $\left(1, \frac{\pi}{2}\right)$ and satisfies the differential equation $\left(7 x^4 \cot y-\mathrm{e}^x \operatorname{cosec} y\right)…
View solution →Let $y=y(x)$ be the solution curve of the differential equation $x\left(x^2+e^x\right) d y+\left(\mathrm{e}^x(x-2) y-x^3\right) \mathrm{d} x=0, x0$, passing through the point…
View solution →Let y = y(x) be the solution of the differential equation $(x^2 + 1)y' - 2xy = (x^4 + 2x^2 + 1)\cos x$, $y(0) = 1$. Then $ \int\limits_{-3}^{3} y(x) \, dx $ is :
View solution →Let $f(x) = x - 1$ and $g(x) = e^x$ for $x \in \mathbb{R}$. If $\frac{dy}{dx} = \left( e^{-2\sqrt{x}} g\left(f(f(x))\right) - \frac{y}{\sqrt{x}} \right)$, $y(0) = 0$, then $y(1)$…
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