Differential Equations · JEE Main previous year questions

Let $y=y(x)$ be the solution of the differential equation $\left(x^2-x\sqrt{x^2-1}\right)dy+\left(y\left(x-\sqrt{x^2-1}\right)-x\right)dx=0$, $x\ge 1$. If $y(1)=1$, then the greatest integer less than $y(\sqrt{5})$ is ____.

Let $y=y(x)$ be the solution of the differential equation $$(\tan x{)}^{1/2}\mathrm{d}y=\left({\sec }^{3}x-(\tan x{)}^{3/2}y\right)\mathrm{d}x,0<x<\frac{\pi }{2},y\left(\frac{\pi }{4}\right)=\frac{6\sqrt{2}}{5}$$ If $y\left(\frac{\pi }{3}\right)=\frac{4}{5}\alpha$, then ${\alpha }^4$ equals

Let $y=y(x)$ be the solution of the differential equation $\frac{dy}{dx}=(1+x+x^2)(1-y+y^2)$, $y(0)=\frac{1}{2}$. Then $(2y(1)-1)$ is equal to

1. A.$\sqrt{3}\tan \left(\frac{11\sqrt{3}}{6}\right)$
2. B.$\frac{\sqrt{3}}{2}\tan \left(\frac{11\sqrt{3}}{12}\right)$
3. C.$\sqrt{3}\tan \left(\frac{11\sqrt{3}}{12}\right)$
4. D.$\frac{\sqrt{3}}{2}\tan \left(\frac{11\sqrt{3}}{6}\right)$

Let $y=y(x)$ be the solution of the differential equation: $$\frac{\mathrm{d}y}{\mathrm{d}x}+\left(\frac{6x^2+\left(3x^2+2x^3+4\right)e^{-2x}}{(x^3+2)\left(2+e^{-2x}\right)}\right)y=2+e^{-2x},$$ $x\in (-1,2)$, satisfying $y(0)=\frac{3}{2}$. If $y(1)=\alpha (2+e^{-2})$, then $\alpha$ is equal to:

1. A.$\frac{13}{8}$
2. B.$\frac{6}{13}$
3. C.$\frac{12}{13}$
4. D.$\frac{13}{12}$

If the curve $y=f(x)$ passes through the point $(1,\mathrm{e})$ and satisfies the differential equation $\mathrm{d}y=y\left(2+{\log }_{\mathrm{e}}x\right)\mathrm{d}x,x>0$, then $f(\mathrm{e})$ is equal to :

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

Let $y=y(x)$ be the solution curve of the differential equation $(1+\sin x)\frac{\mathrm{d}y}{\mathrm{d}x}+(y+1)\cos x=0,y(0)=0$. If the curve $y=y(x)$ passes through the point $\left(\alpha ,\frac{-1}{2}\right)$, then a value of $\alpha$ is :

1. A.$\frac{\pi }{6}$
2. B.$\frac{\pi }{4}$
3. C.$\frac{\pi }{3}$
4. D.$\frac{\pi }{2}$

Let $y=y(x)$ be the solution of the differential equation $x\frac{dy}{dx}-\sin 2y=x^3\left(2-x^3\right){\cos }^{2}y,x\ne 0$. If $y(2)=0$, then $\tan (y(1))$ is equal to

1. A.$-\frac{7}{4}$
2. B.$-\frac{3}{4}$
3. C.$\frac{3}{4}$
4. D.$\frac{7}{4}$

Let $y=y(x)$ be the solution of the differential equation $x\frac{dy}{dx}-y=x^2\cot x,x\in (0,\pi )$. If $y\left(\frac{\pi }{2}\right)=\frac{\pi }{2}$, then $6y\left(\frac{\pi }{6}\right)-8y\left(\frac{\pi }{4}\right)$ is equal to :

1. A.$-3\pi$
2. B.$3\pi$
3. C.$-\pi$
4. D.$\pi$

If the solution curve $y=f(x)$ of the differential equation $$(x^2-4)y^{\prime }-2xy+2x(4-x^2{)}^2=0,x>2,$$ passes through the point $(3,15)$, then the local maximum value of $f$ is _____.

Let $y=y(x)$ be the solution curve of the differential equation $(1+x^2)dy+(y-{\tan }^{-1}x)dx=0,y(0)=1$. Then the value of $y(1)$ is :

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

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 $y(1)$ is

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

If a curve $y=y(x)$ passes through the point $\left(1,\frac{\pi }{2}\right)$ and satisfies the differential equation $\left(7x^4\cot y-e^x\mathrm{cosec}y\right)\frac{dx}{dy}=x^5,x\ge 1$, then at $x=2$, the value of $\cos y$ is :

1. A.$\frac{2e^2+e}{64}$
2. B.$\frac{2e^2-e}{64}$
3. C.$\frac{2e^2-e}{128}$
4. D.$\frac{2e^2+e}{128}$

Let $g$ be a differentiable function such that ${\int }_0^{x}g(t)dt=x-{\int }_0^{x}tg(t)dt$, $x\ge 0$ and let $y=y(x)$ satisfy the differential equation $\frac{dy}{dx}-y\tan x=2(x+1)\sec xg(x)$, $x\in \left[0,\frac{\pi }{2}\right)$. If $y(0)=0$, then $y\left(\frac{\pi }{3}\right)$ is equal to

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

Let $y=y(x)$ be the solution of the differential equation $\frac{dy}{dx}+3\left({\tan }^{2}x\right)y+3y={\sec }^{2}x,y(0)=\frac{1}{3}+e^3$. Then $y\left(\frac{\pi }{4}\right)$ is equal to :

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

Let $y=y(x)$ be the solution of the differential equation $\frac{dy}{dx}+2y{\sec }^{2}x=2{\sec }^{2}x+3\tan x\cdot {\sec }^{2}x$ such that $y(0)=\frac{5}{4}$. Then $12\left(y\left(\frac{\pi }{4}\right)-e^{-2}\right)$ is equal to ____.

If for the solution curve $y=f(x)$ of the differential equation $\frac{dy}{dx}+(\tan x)y=\frac{2+\sec x}{(1+2\sec x{)}^2}$ , $x\in \left(-\frac{\pi }{2},\frac{\pi }{2}\right)$, $f\left(\frac{\pi }{3}\right)=\frac{\sqrt{3}}{10}$, then $f\left(\frac{\pi }{4}\right)$ is equal to:

1. A.$\frac{5-\sqrt{3}}{2\sqrt{2}}$
2. B.$\frac{4-\sqrt{2}}{14}$
3. C.$\frac{9\sqrt{3}+3}{10(4+\sqrt{3})}$
4. D.$-\frac{\sqrt{3}+1}{10(4+\sqrt{3})}$

Let $y=y(x)$ be the solution of the differential equation $2\cos x\frac{dy}{dx}=\sin 2x-4y\sin x,x\in \left(0,\frac{\pi }{2}\right)$. If $y\left(\frac{\pi }{3}\right)=0$, then $y^{\prime }\left(\frac{\pi }{4}\right)+y\left(\frac{\pi }{4}\right)$ is equal to ________.

Let $x=x(y)$ be the solution of the differential equation $y=\left(x-y\frac{\mathrm{d}x}{\mathrm{d}y}\right)\sin \left(\frac{x}{y}\right),y>0$ and $x(1)=\frac{\pi }{2}$. Then $\cos (x(2))$ is equal to :

1. A.$2({\log }_{\mathrm{e}}2)-1$
2. B.$1-2({\log }_{\mathrm{e}}2{)}^2$
3. C.$1-2({\log }_{\mathrm{e}}2)$
4. D.$2({\log }_{\mathrm{e}}2{)}^2-1$

The solution curve, of the differential equation $2y\frac{\mathrm{d}y}{\mathrm{d}x}+3=5\frac{\mathrm{d}y}{\mathrm{d}x}$, passing through the point $(0,1)$ is a conic, whose vertex lies on the line :

1. A.$2x+3y=-9$
2. B.$2x+3y=-6$
3. C.$2x+3y=9$
4. D.$2x+3y=6$

Let $f(x)$ be a positive function such that the area bounded by $y=f(x),y=0$ from $x=0$ to $x=a>0$ is $e^{-a}+4a^2+a-1$. Then the differential equation, whose general solution is $y=c_{1}f(x)+c_2$, where $c_1$ and $c_2$ are arbitrary constants, is

1. A.$(8e^x+1)\frac{d^{2}y}{d{x}^2}-\frac{dy}{dx}=0$
2. B.$(8e^x+1)\frac{d^{2}y}{d{x}^2}+\frac{dy}{dx}=0$
3. C.$(8e^x-1)\frac{d^{2}y}{d{x}^2}-\frac{dy}{dx}=0$
4. D.$(8e^x-1)\frac{d^{2}y}{d{x}^2}+\frac{dy}{dx}=0$

Let $y=y(x)$ be the solution of the differential equation $(1+y^2)e^{\tan x}dx+{\cos }^{2}x(1+e^{2\tan x})dy=0,y(0)=1$. Then $y\left(\frac{\pi }{4}\right)$ is equal to

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

Let $y=y(x)$ be the solution curve of the differential equation $\sec y\frac{\mathrm{d}y}{\mathrm{d}x}+2x\sin y=x^3\cos y,y(1)=0$. Then $y(\sqrt{3})$ is equal to:

1. A.$\frac{\pi }{6}$
2. B.$\frac{\pi }{12}$
3. C.$\frac{\pi }{3}$
4. D.$\frac{\pi }{4}$

If $y(\theta )=\frac{2\cos \theta +\cos 2\theta }{\cos 3\theta +4\cos 2\theta +5\cos \theta +2}$, then at $\theta =\frac{\pi }{2},y^{\prime \prime }+y^{\prime }+y$ is equal to :

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

Let $y=y(x)$ be the solution of the differential equation $\frac{dy}{dx}+\frac{2x}{(1+x^2{)}^2}y=x{e}^{\frac{1}{(1+x^2)}};y(0)=0$. Then the area enclosed by the curve $f(x)=y(x)e^{-\frac{1}{(1+x^2)}}$ and the line $y-x=4$ is _________.

If the solution $y=y(x)$ of the differential equation $(x^4+2x^3+3x^2+2x+2)dy-(2x^2+2x+3)dx=0$ satisfies $y(-1)=-\frac{\pi }{4}$, then $y(0)$ is equal to :

1. A.$-\frac{\pi }{12}$
2. B.$\frac{\pi }{2}$
3. C.0
4. D.$\frac{\pi }{4}$