Differential Equations · JEE Main previous year questions

Let y = y(x) be the solution of the differential equation $\frac{dy}{dx}=1+x{e}^{y-x},-\sqrt{2}<x<\sqrt{2},y(0)=0$ then, the minimum value of $y(x),x\in \left(-\sqrt{2},\sqrt{2}\right)$ is equal to :

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

Let y = y(x) be solution of the following differential equation $e^y\frac{dy}{dx}-2e^y\sin x+\sin x{\cos }^{2}x=0,y\left(\frac{\pi }{2}\right)=0$ If $y(0)={\log }_e(\alpha +\beta e^{-2})$, then $4(\alpha +\beta )$ is equal to ____.

The differential equation satisfied by the system of parabolas $y^2=4a(x+a)$ is :

1. A.$y{\left(\frac{dy}{dx}\right)}^2-2x\left(\frac{dy}{dx}\right)-y=0$
2. B.$y{\left(\frac{dy}{dx}\right)}^2-2x\left(\frac{dy}{dx}\right)+y=0$
3. C.$y{\left(\frac{dy}{dx}\right)}^2+2x\left(\frac{dy}{dx}\right)-y=0$
4. D.$y\left(\frac{dy}{dx}\right)+2x\left(\frac{dy}{dx}\right)-y=0$

Let y = y(x) be the solution of the differential equation $\frac{dy}{dx}=(y+1)\left((y+1)e^{x^2/2}-x\right),0<x<2.1$, with $y(2)=0$. Then the value of $\frac{dy}{dx}$ at $x=1$ is equal to :

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

If $y=y(x)$ is the solution of the differential equation, $\frac{dy}{dx}+2y\tan x=\sin x$, $y\left(\frac{\pi }{3}\right)=0$, then the maximum value of the function $y(x)$ over $\mathbb{R}$ is equal to:

1. A.$\frac{1}{8}$
2. B.8
3. C.$-\frac{15}{4}$
4. D.$\frac{1}{2}$

Let the curve $y=y(x)$ be the solution of the differential equation, $\frac{dy}{dx}=2(x+1)$. If the numerical value of area bounded by the curve $y=y(x)$ and x-axis is $\frac{4\sqrt{8}}{3}$, then the value of y(1) is equal to _________.

The difference between degree and order of a differential equation that represents the family of curves given by $y^2=a\left(x+\frac{\sqrt{a}}{2}\right),a>0$ is _________.

If $y=y(x)$ is the solution of the equation $e^{\sin y}\cos y\frac{dy}{dx}+e^{\sin y}\cos x=\cos x$, $y(0)=0$; then $1+y\left(\frac{\pi }{6}\right)+\frac{\sqrt{3}}{2}y\left(\frac{\pi }{3}\right)+\frac{1}{\sqrt{2}}y\left(\frac{\pi }{4}\right)$ is equal to _________.

If $y=\left(\frac{2}{\pi }x-1\right)\mathrm{cosec}x$ is the solution of the differential equation, $$\frac{dy}{dx}+p(x)y=\frac{2}{\pi }\mathrm{cosec}x,$$ $0<x<\frac{\pi }{2}$, then the function $p(x)$ is equal to :

1. A.$\cot x$
2. B.$\sec x$
3. C.$\tan x$
4. D.$\mathrm{cosec}x$

The solution curve of the differential equation, $(1+e^{-x})(1+y^2)\frac{dy}{dx}=y^2$, which passes through the point $(0,1)$, is :

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

Let $y=y(x)$ be the solution of the differential equation, $\frac{2+\sin x}{y+1}\cdot \frac{dy}{dx}=-\cos x$, $y>0,y(0)=1$. If $y(\pi )=a$ and $\frac{dy}{dx}$ at $x=\pi$ is b, then the ordered pair (a, b) is equal to :

1. A.$(2,1)$
2. B.$\left(2,\frac{3}{2}\right)$
3. C.$(1,-1)$
4. D.$(1,1)$

Let $y=y(x)$ be a solution of the differential equation, $$\sqrt{1-x^2}\frac{dy}{dx}+\sqrt{1-y^2}=0,|x|<1.$$ If $y\left(\frac{1}{2}\right)=\frac{\sqrt{3}}{2}$, then $y\left(-\frac{1}{\sqrt{2}}\right)$ is equal to :

1. A.$-\frac{\sqrt{3}}{2}$
2. B.None of those
3. C.$\frac{1}{\sqrt{2}}$
4. D.$-\frac{1}{\sqrt{2}}$

The differential equation of the family of curves, $x^2=4b(y+b),b\in \mathbf{R}$, is :

1. A.$x(y^{\prime }{)}^2=x-2y{y}^{\prime }$
2. B.$x(y^{\prime }{)}^2=2y{y}^{\prime }-x$
3. C.$x{y}^{\prime \prime }=y^{\prime }$
4. D.$x(y^{\prime }{)}^2=x+2y{y}^{\prime }$

If $y=y(x)$ is the solution of the differential equation, $e^y\left(\frac{dy}{dx}-1\right)=e^x$ such that $y(0)=0$, then $y(1)$ is equal to:

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

Let $y=y(x)$ be the solution curve of the differential equation, $(y^2-x)\frac{dy}{dx}=1$, satisfying $y(0)=1$. This curve intersects the x-axis at a point whose abscissa is :

1. A.$2+e$
2. B.$-e$
3. C.2
4. D.$2-e$

Let $z\in \mathbb{C}$ with $\text{Im}(z)=10$ and it satisfies $\frac{2z-n}{2z+n}=2i-1$ for some natural number n. Then :

1. A.$n=20$ and $\text{Re}(z)=-10$
2. B.$n=40$ and $\text{Re}(z)=10$
3. C.$n=40$ and $\text{Re}(z)=-10$
4. D.$n=20$ and $\text{Re}(z)=10$

If y = y(x) is the solution of the differential equation $\frac{dy}{dx}=(\tan x-y){\sec }^{2}x$, $x\in \left(-\frac{\pi }{2},\frac{\pi }{2}\right)$, such that y(0) = 0, then $y\left(-\frac{\pi }{4}\right)$ is equal to :

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

Let y = y(x) be the solution of the differential equation, $\frac{dy}{dx}+y\tan x=2x+x^2\tan x$, $x\in \left(-\frac{\pi }{2},\frac{\pi }{2}\right)$, such that y(0) = 1. Then :

1. A.$y\left(\frac{\pi }{4}\right)-y\left(-\frac{\pi }{4}\right)=\sqrt{2}$
2. B.$y^{\prime }\left(\frac{\pi }{4}\right)-y^{\prime }\left(-\frac{\pi }{4}\right)=\pi -\sqrt{2}$
3. C.$y\left(\frac{\pi }{4}\right)+y\left(-\frac{\pi }{4}\right)=\frac{{\pi }^2}{2}+2$
4. D.$y^{\prime }\left(\frac{\pi }{4}\right)+y^{\prime }\left(-\frac{\pi }{4}\right)=-\sqrt{2}$

The solution of the differential equation $x\frac{dy}{dx}+2y=x^2(x\ne 0)$ with y(1) = 1, is :

1. A.$y=\frac{4}{5}{x}^3+\frac{1}{5x^2}$
2. B.$y=\frac{3}{4}{x}^2+\frac{1}{4x^2}$
3. C.$y=\frac{x^2}{4}+\frac{3}{4x^2}$
4. D.$y=\frac{x^3}{5}+\frac{1}{5x^2}$

Let $y=y(x)$ be the solution of the differential equation, $x\frac{dy}{dx}+y=x{\log }_{e}x$, $(x>1)$. If $2y(2)={\log }_{e}4-1$, then $y(e)$ is equal to :

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

The solution of the differential equation, $$\frac{dy}{dx}=(x-y{)}^2,\text{ when }y(1)=1,\text{ is :}$$

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

If $y=y(x)$ is the solution of the differential equation, $$x\frac{dy}{dx}+2y=x^2,\text{ satisfying }y(1)=1,\text{ then }y\left(\frac{1}{2}\right)\text{ is equal to :}$$

1. A.$\frac{7}{64}$
2. B.$\frac{49}{16}$
3. C.$\frac{1}{4}$
4. D.$\frac{13}{16}$

Let $y=y(x)$ be the solution of the differential equation $\frac{dy}{dx}+2y=f(x)$, where $f(x)=\{\begin{array}{ll}1,& x\in [0,1]\\ 0,& \text{otherwise}\end{array}$ If $y(0)=0$, then $y\left(\frac{3}{2}\right)$ is :

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

The curve satisfying the differential equation, $y\mathrm{d}x-(x+3y^2)\mathrm{d}y=0$ and passing through the point $(1,1)$, also passes through the point :

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

If a curve $y=f(x)$ passes through the point $(1,-1)$ and satisfies the differential equation, $y(1+xy)dx=xdy$, then $f\left(-\frac{1}{2}\right)$ is equal to :

1. A.$\frac{2}{5}$
2. B.$\frac{4}{5}$
3. C.$-\frac{2}{5}$
4. D.$-\frac{4}{5}$