Differential Equations · JEE Main previous year questions

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 ____.

Let $y=y(x)$ be the solution of the differential equation $(x^2+4{)}^{2}dy+(2x^{3}y+8xy-2)dx=0$. If $y(0)=0$, then $y(2)$ is equal to

1. A.$2\pi$
2. B.$\frac{\pi }{8}$
3. C.$\frac{\pi }{16}$
4. D.$\frac{\pi }{32}$

Let $y=y(x)$ be the solution of the differential equation $(x+y+2{)}^{2}dx=dy,y(0)=-2$. Let the maximum and minimum values of the function $y=y(x)$ in $\left[0,\frac{\pi }{3}\right]$ be $\alpha$ and $\beta$, respectively. If $(3\alpha +\pi {)}^2+{\beta }^2=\gamma +\delta \sqrt{3},\gamma ,\delta \in \mathbb{Z}$, then $\gamma +\delta$ equals _________.

Let $y=y(x)$ be the solution of the differential equation $\frac{dy}{dx}=2x(x+y{)}^3-x(x+y)-1,y(0)=1$. Then, ${\left(\frac{1}{\sqrt{2}}+y\left(\frac{1}{\sqrt{2}}\right)\right)}^2$ equals :

1. A.$\frac{4}{4+\sqrt{e}}$
2. B.$\frac{3}{3-\sqrt{e}}$
3. C.$\frac{2}{1+\sqrt{e}}$
4. D.$\frac{1}{2-\sqrt{e}}$

Let $y=y(x)$ be the solution of the differential equation $\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{(\tan x)+y}{\sin x(\sec x-\sin x\tan x)},x\in \left(0,\frac{\pi }{2}\right)$ satisfying the condition $y\left(\frac{\pi }{4}\right)=2$. Then, $y\left(\frac{\pi }{3}\right)$ is

1. A.$\sqrt{3}\left(2+{\log }_{e}3\right)$
2. B.$\sqrt{3}\left(1+2{\log }_{e}3\right)$
3. C.$\sqrt{3}\left(2+{\log }_e\sqrt{3}\right)$
4. D.$\frac{\sqrt{3}}{2}\left(2+{\log }_{e}3\right)$

The solution curve of the differential equation $y\frac{\mathrm{d}x}{\mathrm{d}y}=x\left({\log }_{e}x-{\log }_{e}y+1\right),x>0,y>0$ passing through the point $(e,1)$ is

1. A.$\left|{\log }_e\frac{y}{x}\right|=y^2$
2. B.$\left|{\log }_e\frac{y}{x}\right|=x$
3. C.$\left|{\log }_e\frac{x}{y}\right|=y$
4. D.$2\left|{\log }_e\frac{x}{y}\right|=y+1$

Let $y=y(x)$ be the solution of the differential equation $${\sec }^{2}xdx+\left(e^{2y}{\tan }^{2}x+\tan x\right)dy=0,0<x<\frac{\pi }{2},y(\pi /4)=0.$$ If $y(\pi /6)=\alpha$, then $e^{8\alpha }$ is equal to ____.

Let $y=y(x)$ be the solution of the differential equation $\sec xdy+\{2(1-x)\tan x+x(2-x)\}dx=0$ such that $y(0)=2$. Then $y(2)$ is equal to:

1. A.$2\{\sin (2)+1\}$
2. B.2
3. C.1
4. D.$2\{1-\sin (2)\}$

Let $y=y(x)$ be the solution of the differential equation $(1-x^2)dy=\left[xy+(x^3+2)\sqrt{3(1-x^2)}\right]dx,-1<x<1,y(0)=0$. If $y\left(\frac{1}{2}\right)=\frac{m}{n}$, m and n are co-prime numbers, then $m+n$ is equal to _________.

If the solution curve $y=y(x)$ of the differential equation $(1+y^2)(1+{\log }_{e}x)\mathrm{d}x+x\mathrm{d}y=0$, $x>0$ passes through the point $(1,1)$ and $y(e)=\frac{\alpha -\tan \left(\frac{3}{2}\right)}{\beta +\tan \left(\frac{3}{2}\right)}$, then $\alpha +2\beta$ is _______.

If $\sin \left(\frac{y}{x}\right)={\log }_e|x|+\frac{\alpha }{2}$ is the solution of the differential equation $x\cos \left(\frac{y}{x}\right)\frac{dy}{dx}=y\cos \left(\frac{y}{x}\right)+x$ and $y(1)=\frac{\pi }{3}$, then ${\alpha }^2$ is equal to

1. A.12
2. B.9
3. C.4
4. D.3

Let $x=x(t)$ and $y=y(t)$ be solutions of the differential equations $\frac{\mathrm{d}x}{\mathrm{d}t}+\mathrm{a}x=0$ and $\frac{\mathrm{d}y}{\mathrm{d}t}+\mathrm{b}y=0$ respectively, $\mathrm{a},\mathrm{b}\in \mathbf{R}$. Given that $x(0)=2$; $y(0)=1$ and $3y(1)=2x(1)$, the value of $\mathrm{t}$, for which $x(t)=y(t)$, is :

1. A.${\log }_{\frac{2}{3}}2$
2. B.${\log }_{\frac{4}{3}}2$
3. C.${\log }_{4}3$
4. D.${\log }_{3}4$

If the solution of the differential equation $(2x+3y-2)\mathrm{d}x+(4x+6y-7)\mathrm{d}y=0$, $y(0)=3$, is $\alpha x+\beta y+3{\log }_e|2x+3y-\gamma |=6$, then $\alpha +2\beta +3\gamma$ is equal to ____.

If $y=y(x)$ is the solution curve of the differential equation $(x^2-4)dy-(y^2-3y)dx=0$, $x>2,y(4)=\frac{3}{2}$ and the slope of the curve is never zero, then the value of $y(10)$ equals :

1. A.$\frac{3}{1+(8{)}^{1/4}}$
2. B.$\frac{3}{1-(8{)}^{1/4}}$
3. C.$\frac{3}{1-2\sqrt{2}}$
4. D.$\frac{3}{1+2\sqrt{2}}$

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 ${\tan }^{-1}\left(\frac{y-1}{x-1}\right)-\frac{1}{\beta }{\log }_e\left(\alpha +{\left(\frac{y-1}{x-1}\right)}^2\right)={\log }_e|x-1|$, then $5\beta +\alpha$ is equal to _______.

Let $y=y_1(x)$ and $y=y_2(x)$ be the solution curves of the differential equation $\frac{dy}{dx}=y+7$ with initial conditions $y_1(0)=0$ and $y_2(0)=1$ respectively. Then the curves $y=y_1(x)$ and $y=y_2(x)$ intersect at

1. A.no point
2. B.two points
3. C.infinite number of points
4. D.one point

If $y=y(x)$ is the solution of the differential equation $$\frac{dy}{dx}+\frac{4x}{(x^2-1)}y=\frac{x+2}{(x^2-1{)}^{\frac{5}{2}}},x>1$$ such that $y(2)=\frac{2}{9}{\log }_e(2+\sqrt{3})$ and $y(\sqrt{2})=\alpha {\log }_e(\sqrt{\alpha }+\beta )+\beta -\sqrt{\gamma }$, $\alpha ,\beta ,\gamma \in \mathbb{N}$, then $\alpha \beta \gamma$ is equal to :

Let $y=y(x)$, $y>0$, be a solution curve of the differential equation $(1+x^2)dy=y(x-y)dx$. If $y(0)=1$ and $y(2\sqrt{2})=\beta$, then

1. A.$e^{{\beta }^{-1}}=e^{-2}(3+2\sqrt{2})$
2. B.$e^{3{\beta }^{-1}}=e(5+\sqrt{2})$
3. C.$e^{3{\beta }^{-1}}=e(3+2\sqrt{2})$
4. D.$e^{{\beta }^{-1}}=e^{-2}(5+\sqrt{2})$

Let $y=y(x)$ be a solution curve of the differential equation $(1-x^2{y}^2)dx=ydx+xdy$. If the line $x=1$ intersects the curve $y=y(x)$ at $y=2$ and the line $x=2$ intersects the curve $y=y(x)$ at $y=\alpha$, then a value of $\alpha$ is :

1. A.$\frac{1+3e^2}{2(3e^2-1)}$
2. B.$\frac{3e^2}{2(3e^2-1)}$
3. C.$\frac{1-3e^2}{2(3e^2+1)}$
4. D.$\frac{3e^2}{2(3e^2+1)}$

Let $y=y(x)$ be the solution of the differential equation $\frac{dy}{dx}+\frac{5}{x(x^5+1)}y=\frac{(x^5+1{)}^2}{x^7}$, $x>0$. If $y(1)=2$, then $y(2)$ is equal to :

1. A.$\frac{693}{128}$
2. B.$\frac{697}{128}$
3. C.$\frac{637}{128}$
4. D.$\frac{679}{128}$

Let the tangent at any point P on a curve passing through the points $(1,1)$ and $\left(\frac{1}{10},100\right)$, intersect positive x-axis and y-axis at the points A and B respectively. If $\text{PA}:\text{PB}=1:k$ and $y=y(x)$ is the solution of the differential equation $e^{\frac{dy}{dx}}=kx+\frac{k}{2},y(0)=k$, then $4y(1)-6{\log }_{e}3$ is equal to _________.

If the solution curve of the differential equation $(y-2{\log }_{e}x)dx+(x{\log }_e{x}^2)dy=0,x>1$ passes through the points $\left(e,\frac{4}{3}\right)$ and $(e^4,\alpha )$, then $\alpha$ is equal to ___________.

Let the solution curve $x=x(y),0<y<\frac{\pi }{2}$, of the differential equation $({\log }_e(\cos y){)}^2\cos ydx-(1+3x{\log }_e(\cos y))\sin ydy=0$ satisfy $x\left(\frac{\pi }{3}\right)=\frac{1}{2{\log }_{e}2}$. If $x\left(\frac{\pi }{6}\right)=\frac{1}{{\log }_{e}m-{\log }_{e}n}$, where $m$ and $n$ are coprime, then $mn$ is equal to ___________.

If the solution curve $f(x,y)=0$ of the differential equation $$(1+{\log }_{e}x)\frac{dx}{dy}-x{\log }_{e}x=e^y,x>0,$$ passes through the points $(1,0)$ and $(\alpha ,2)$, then ${\alpha }^{\alpha }$ is equal to :

1. A.$e^{\sqrt{2}{e}^2}$
2. B.$e^{2e^{\sqrt{2}}}$
3. C.$e^{e^2}$
4. D.$e^{2e^2}$

The area enclosed by the closed curve $C$ given by the differential equation $$\frac{dy}{dx}+\frac{x+a}{y-2}=0,y(1)=0\text{ is }4\pi .$$ Let $P$ and $Q$ be the points of intersection of the curve $C$ and the $y$-axis. If normals at $P$ and $Q$ on the curve $C$ intersect $x$-axis at points $R$ and $S$ respectively, then the length of the line segment $RS$ is :

1. A.$\frac{4\sqrt{3}}{3}$
2. B.$2\sqrt{3}$
3. C.$2$
4. D.$\frac{2\sqrt{3}}{3}$