Question

Let $y=y(x)$ be the solution of the differential equation $\left(x^2+4\right)^2 d y+\left(2 x^3 y+8 x y-2\right) d x=0$. If $y(0)=0$, then $y(2)$ is equal to

JEE Main 2024 (04 Apr Shift 2)

Options

• A: $\frac{\pi}{32}$
• B: ${2}{\pi}$
• C: $\frac{\pi}{8}$
• D: $\frac{\pi}{16}$

Question

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 :

JEE Main 2024 (08 Apr Shift 2)

Options

• A: $\frac{\pi}{3}$
• B: $\frac{\pi}{6}$
• C: $\frac{\pi}{12}$
• D: $\frac{\pi}{4}$

Question

Let $y=y\left(x\right)$ be the solution of the differential equation $\left(x-x^3\right)dy=\left(y+y{x}^2-3x^4\right)dx,x>2$ If $y\left(3\right)=3,$ then $y\left(4\right)$ is equal to:

JEE Main 2021 (27 Jul Shift 2)

Options

• A: $4$
• B: $12$
• C: $8$
• D: $16$

Question

If $y=y\left(x\right),x\in \left(0,\frac{\pi }{2}\right)$ be the solution curve of the differential equation

$\left({\sin }^{2}2x\right)\frac{dy}{dx}+\left(8{\sin }^{2}2x+2\sin 4x\right)y=$

$2e^{-4x}\left(2\sin 2x+\cos 2x\right),$ with $y\left(\frac{\pi }{4}\right)=e^{-\pi }$, then $y\left(\frac{\pi }{6}\right)$ is equal to

JEE Main 2022 (28 Jul Shift 1)

Options

• A: $\frac{2}{\sqrt{3}}{e}^{-\frac{2\pi }{3}}$
• B: $\frac{2}{\sqrt{3}}{e}^{\frac{2\pi }{3}}$
• C: $\frac{1}{\sqrt{3}}{e}^{-\frac{2\pi }{3}}$
• D: $\frac{1}{\sqrt{3}}{e}^{\frac{2\pi }{3}}$

Question

Suppose $y=y\left(x\right)$ be the solution curve to the differential equation $\frac{dy}{dx}-y=2-e^{-x}$ such that $\underset{x\to \infty }{\lim }y\left(x\right)$ is finite. If $a$ and $b$ are respectively the $x-$ and $y-$intercept of the tangent to the curve at $x=0$, then the value of $a-4b$ is equal to _______.

JEE Main 2022 (26 Jul Shift 2)

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Question

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

JEE Main 2023 (11 Apr Shift 1)

Options

• A: $\frac{1-3e^2}{2\left(3e^2+1\right)}$
• B: $\frac{1+3e^2}{2\left(3e^2-1\right)}$
• C: $\frac{3e^2}{2\left(3e^2-1\right)}$
• D: $\frac{3e^2}{2\left(3e^2+1\right)}$

Question

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

JEE Main 2024 (30 Jan Shift 1)

Options

• A: $2$
• B: $2\{1-\sin (2\left)\right\}$
• C: $2\left\{\sin \right(2)+1\}$
• D: $1$

Question

Let $y=y\left(x\right)$ satisfies the equation $\frac{dy}{dx}-\left|A\right|=0,$ for all $x>0,$ where $A=\left[\begin{array}{ccc}y& \sin x& 1\\ 0& -1& 1\\ 2& 0& \frac{1}{x}\end{array}\right]$. If $y\left(\pi \right)=\pi +2,$ then the value of $y\left(\frac{\pi }{2}\right)$ is:

JEE Main 2021 (20 Jul Shift 2)

Options

• A: $\frac{\pi }{2}+\frac{4}{\pi }$
• B: $\frac{\pi }{2}-\frac{1}{\pi }$
• C: $\frac{3\pi }{2}-\frac{1}{\pi }$
• D: $\frac{\pi }{2}-\frac{4}{\pi }$

Question

Let the solution curve $y=y\left(x\right)$ of the differential equation, $\left[\frac{x}{\sqrt{x^2-y^2}}+e^{\frac{y}{x}}\right]x\frac{dy}{dx}=x+\left[\frac{x}{\sqrt{x^2-y^2}}+e^{\frac{y}{x}}\right]y$ pass through the points $\left(1,0\right)$ and $\left(2\alpha ,\alpha \right),\alpha >0$. Then $\alpha$ is equal to

JEE Main 2022 (28 Jun Shift 1)

Options

• A: $\frac{1}{2}\exp \left(\frac{\pi }{6}+\sqrt{\mathrm{e}}-1\right)$
• B: $\frac{1}{2}\exp \left(\frac{\pi }{3}+\sqrt{e}-1\right)$
• C: $\exp \left(\frac{\pi }{6}+\sqrt{\mathrm{e}}+1\right)$
• D: $2\exp \left(\frac{\pi }{3}+\sqrt{\mathrm{e}}-1\right)$

Question

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

JEE Main 2023 (06 Apr Shift 2)

Options

• A: ${e^{2e}}^2$
• B: $e^{e^2}$
• C: ${e^{\sqrt{2e}}}^2$
• D: $e^{2e\sqrt{2}}$

Question

If $y=y\left(x\right)$ is the solution of the differential equation $2x^2\frac{dy}{dx}-2xy+3y^2=0$ such that $y\left(e\right)=\frac{e}{3}$, then $y\left(1\right)$ is equal to

JEE Main 2022 (25 Jun Shift 2)

Options

• A: $\frac{1}{3}$
• B: $\frac{2}{3}$
• C: $\frac{3}{2}$
• D: $3$

Question

Let $y=y\left(x\right)$ be the solution of the differential equation $\frac{dy}{dx}=\frac{4y^3+2y{x}^2}{3x{y}^2+x^3},y\left(1\right)=1$. If for some $n\in N,y\left(2\right)\in [n-1,n)$, then $n$ is equal to _______.

JEE Main 2022 (25 Jul Shift 2)

Enter your answer

Question

A function $y=f\left(x\right)$ satisfies $f\left(x\right)\sin 2x+\sin x-\left(1+{\cos }^{2}x\right)f^{\text{'}}\left(x\right)=0$ with condition $f\left(0\right)=0$. Then $f\left(\frac{\pi }{2}\right)$ is equal to

JEE Main 2024 (29 Jan Shift 1)

Options

• A: $1$
• B: $0$
• C: $-1$
• D: $2$

Question

Let $y=y\left(x\right)$ be the solution of the differential equation $\left(x+1\right)y^{\text{'}}-y={\mathrm{e}}^{3x}{\left(x+1\right)}^2$, with $y\left(0\right)=\frac{1}{3}$. Then, the point $x=-\frac{4}{3}$ for the curve $y=y\left(x\right)$ is

JEE Main 2022 (25 Jun Shift 1)

Options

• A: not a critical point
• B: a point of local minima
• C: a point of local maxima
• D: a point of inflection

Question

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, whose vertex lies on the line:

JEE Main 2024 (09 Apr Shift 1)

Options

• A: $2 x+3 y=9$
• B: $2 x+3 y=-9$
• C: $2 x+3 y=-6$
• D: $2 x+3 y=6$

Question

Let $y=y\left(x\right)$ be the solution curve of the differential equation $\frac{dy}{dx}+\left(\frac{2x^2+11x+13}{x^3+6x^2+11x+6}\right)y=\frac{\left(x+3\right)}{x+1},x>-1$, which passes through the point $\left(0,1\right)$. Then $y\left(1\right)$ is equal to

JEE Main 2022 (29 Jul Shift 2)

Options

• A: $\frac{1}{2}$
• B: $\frac{3}{2}$
• C: $\frac{5}{2}$
• D: $\frac{7}{2}$

Question

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 origin. Then the radius of this circle is :

JEE Main 2024 (06 Apr Shift 2)

Options

• A: 2
• B: $\sqrt{17}$
• C: $\frac{1}{2}$
• D: $\frac{\sqrt{17}}{2}$

Question

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

JEE Main 2022 (28 Jun Shift 2)

Options

• A: $e{\log }_e\left(2\right)$
• B: $-e{\log }_e\left(2\right)$
• C: $e^2{\log }_e\left(2\right)$
• D: $-e^2{\log }_e\left(2\right)$

Question

If the solution curve of the differential equation $\frac{dy}{dx}=\frac{x+y-2}{x-y}$ passes through the point $\left(2,1\right)$ and $\left(k+1,2\right),\mathrm{k}>0$, then

JEE Main 2022 (29 Jul Shift 2)

Options

• A: $2{\tan }^{-1}\left(\frac{1}{k}\right)={\log }_{\mathrm{e}}\left(k^2+1\right)$
• B: ${\tan }^{-1}\left(\frac{1}{k}\right)={\log }_{\mathrm{e}}\left(k^2+1\right)$
• C: $2{\tan }^{-1}\left(\frac{1}{k+1}\right)={\log }_{\mathrm{e}}\left(k^2+2k+2\right)$
• D: $2{\tan }^{-1}\left(\frac{1}{k}\right)={\log }_{\mathrm{e}}\left(\frac{k^2+1}{k^2}\right)$

Question

The slope of normal at any point $\left(x,y\right),x>0,y>0$ on the curve $y=y\left(x\right)$ is given by $\frac{x^2}{xy-x^2{y}^2-1}$. If the curve passes through the point $\left(1,1\right)$, then $e·y\left(e\right)$ is equal to

JEE Main 2022 (24 Jun Shift 2)

Options

• A: $\frac{1-\tan \left(1\right)}{1+\tan \left(1\right)}$
• B: $\tan \left(1\right)$
• C: $1$
• D: $\frac{1+\tan \left(1\right)}{1-\tan \left(1\right)}$

Question

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 $\left[0, \frac{\pi}{3}\right]$ 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 ______

JEE Main 2024 (04 Apr Shift 2)

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Question

If $\frac{\mathrm{d}y}{ \mathrm{d}x}=\frac{2^{x+y}-2^x}{2^y},y\left(0\right)=1,$ then $y\left(1\right)$ is equal to :

JEE Main 2021 (31 Aug Shift 1)

Options

• A: ${\log }_2\left(1+{\mathrm{e}}^2\right)$
• B: ${\log }_2\left(2\mathrm{e}\right)$
• C: ${\log }_2(2+\mathrm{e})$
• D: ${\log }_2(1+e)$

Question

The general solution of the differential equation $\left(x-y^2\right)dx+y\left(5x+y^2\right)dy=0$ is

JEE Main 2022 (25 Jul Shift 1)

Options

• A: ${\left(y^2+x\right)}^4=C\left|{\left(y^2+2x\right)}^3\right|$
• B: ${\left(y^2+2x\right)}^4=C\left|{\left(y^2+x\right)}^3\right|$
• C: $\left|{\left(y^2+x\right)}^3\right|=C{\left(2y^2+x\right)}^4$
• D: $\left|{\left(y^2+2x\right)}^3\right|=C{\left(2y^2+x\right)}^4$

Question

Let $x=x\left(y\right)$ be the solution of the differential equation $2\left(y+2\right){\log }_e\left(y+2\right)dx+\left(x+4-2{\log }_e\left(y+2\right)\right)dy=0$, $y>-1$ with $x\left(e^4-2\right)=1$. Then $x\left(e^9-2\right)$ is equal to

JEE Main 2023 (15 Apr Shift 1)

Options

• A: $3$
• B: $\frac{4}{9}$
• C: $\frac{32}{9}$
• D: $\frac{10}{3}$

Question

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

JEE Main 2024 (04 Apr Shift 1)

Options

• A: $\frac{\pi}{2}$
• B: $-\frac{\pi}{2}$
• C: 0
• D: $\frac{\pi}{4}$