Differential Equations

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

[JEE Main 2024, 29 Jan (Shift 1)]

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

[JEE Main 2024, 29 Jan (Shift 1)]

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

[JEE Main 2026, 28 Jan (Shift 1)]

If \(x=f(y)\) is the solution of the differential equation \( \left(1+y^2\right)+\left(x-2 e^{\tan ^{-1} y}\right) \frac{d y}{d x}=0, y \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \) with \(f(0)=1\), then \(f\left(\frac{1}{\sqrt{3}}\right)\) is equal to :

[JEE Main 2025, 22 Jan (Shift 2)]

Let \(y=y(x)\) be a differentiable function in the interval \((0,\infty )\) such that \(y(1)=2,\) and \(\lim _{t\to x}\frac{{t}^{2}y\left(x\right)−{x}^{2}y\left(t\right)}{x−t}=3\) for each \(x>0.\) Then \(2y(2)\) is equal to

[JEE Main 2026, 24 Jan (Shift 2)]

Let \(\mathrm{x}=\mathrm{x}(\mathrm{y})\) be the solution of the differential equation

\(y=\left(x-y \frac{d x}{d y}\right) \sin \left(\frac{x}{y}\right), y>0 \text { and } x(1)=\frac{\pi}{2} .\)

Then \(\cos (x(2))\) is equal to :

[JEE Main 2025, 23 Jan (Shift 2)]

Let \(y:(-\infty, \infty) \rightarrow(0, \infty)\) be the solution of the differential equation \(\frac{dy}{dx}=\frac{{e}^{5x}{y}^{3}+{y}^{3}}{{e}^{x}+{e}^{x}{y}^{4}}\) satisfying \(y\left(0\right)=\frac{1}{\sqrt{2}}\). Then the value of \(y\left(\log _e 2\right)\) is

[JEE Advanced 2026]

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

[JEE Main 2026, 2 Apr (Shift 1)]

If \(\frac{d y}{d x}+\left(\frac{x}{1+x^2}\right) y=\frac{\sqrt{x}}{\sqrt{1+x^2}} ; y(0)=0\), then \(y(1)\) will be (24 Jan, Shift I, Memory Based)

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

[JEE Main 2026, 4 Apr (Shift 2)]

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

[JEE Main 2024, 8 Apr (Shift 1)]

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

[JEE Main 2024, 8 Apr (Shift 1)]

\(\text { If } 2 \cos x \frac{d y}{d x}=\sin 2 x-4 y \sin x \cdot y\left(\frac{\pi}{3}\right)=0 \text { find } y^{\prime}\left(\frac{\pi}{4}\right)+y\left(\frac{\pi}{4}\right)\).

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

[JEE Main 2024, 1 Feb (Shift 1)]

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

[JEE Main 2025, 3 Apr (Shift 2)]

If \(\frac{d y}{d x}+\left(\frac{x}{1+x^2}\right) y=\frac{\sqrt{x}}{\sqrt{1+x^2}} ; y(0)=0\), then \(y(1)\) will be (24 Jan, Shift I, Memory Based)

The differential equation of the family of circles passing through the origin and having centre at the line \(y=x\) is :

[JEE Main 2024, 5 Apr (Shift 2)]

\(\begin{aligned}&f(y) \text { is the solution of differential equation }\\&\left(1+y^2\right)+\left(x-2 \tan ^{-1} y\right) \frac{d y}{d x}=0, f(0)=1, \text { find } f\left(\frac{1}{\sqrt{3}}\right) .\end{aligned}\) (22 Jan, Shift II, Memory Based)

\(\text { If } y=\left(x-y \frac{d x}{d y}\right) \sin \left(\frac{x}{y}\right) \text { if } x(1)=\frac{\pi}{2} \text { then find } \cos (x(2)) \text {. }\)

If \(\frac{d y}{d x}-y \log _e 2=2^{\sin x}(\cos x-1) \log _e 2\), then \(y\) is:

If \(\sqrt{4-{x}^{2}}\frac{dy}{dx}=({({\sin }^{-1}\frac{x}{2})}^{2}-2){\sin }^{-1}(\frac{x}{2}),\) then \(y(0)=-6,Findy(2):\)

Let \(y=y(x)\) be the solution of the differential equation \(x \sin \left(\frac{y}{x}\right) d y=\left(y \sin \left(\frac{y}{x}\right)-x\right) d x, y(1)=\frac{\pi}{2}\) and let \(\alpha=\cos \left(\frac{y\left(e^{12}\right)}{e^{12}}\right)\). Then the number of integral values of \(p\), for which the equation \(x^2+y^2-2 p x+2 p y+\alpha+2=0\) represents a circle of radius \(r \leq 6\), is \(\_\_\_\_\) .

[JEE Main 2026, 5 Apr (Shift 1)]

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

Let \(\mathrm{y}=\mathrm{y}(\mathrm{x})\) be the solution of the differential equation \(\left({x}^{2}+1\right){y}^{'}-2xy=\left({x}^{4}+2{x}^{2}+1\right)\cos x\), \(y(0)=1\). Then \({\int }_{-3}^{3}y(x)dx\) is :

[JEE Main 2025, 7 Apr (Shift 2)]

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

[JEE Main 2024, 31 Jan (Shift 1)]

The solution of the differential equation \((x+1) \frac{d y}{d x}-y=e^{3 x}(x+1)^2\) is:

Let \(y=y(x)\) be the solution curve of the differential equation \(\mathrm{x}\left({\mathrm{x}}^{2}+{\mathrm{e}}^{\mathrm{x}}\right)\mathrm{dy}+\left({\mathrm{e}}^{\mathrm{x}}(\mathrm{x}-2)\mathrm{y}-{\mathrm{x}}^{3}\right)\mathrm{dx}=0,\) x > 0 passing through the point \((1,0)\). Then \(y(2)\) is equal to :

[JEE Main 2025, 7 Apr (Shift 1)]

If \(\sqrt{4-{x}^{2}}\frac{dy}{dx}=({({\sin }^{-1}\frac{x}{2})}^{2}-2){\sin }^{-1}(\frac{x}{2}),\) then \(y(0)=-6,Findy(2):\)

Let \(y = y(x)\) be the solution of the differential equation \(\left(xy-5{x}^{2}\sqrt{1+{x}^{2}}\right)dx+\left(1+{x}^{2}\right)dy=0\),\(y(0) = 0.\) Then \(y(\sqrt{3})\) is equal to

[JEE Main 2025, 24 Jan (Shift 1)]

\(\text { If } y=\left(x-y \frac{d x}{d y}\right) \sin \left(\frac{x}{y}\right) \text { if } x(1)=\frac{\pi}{2} \text { then find } \cos (x(2)) \text {. }\)

If the curve satisfying the differential equation \(\frac{d y}{d x}=\frac{6-2 e^{2 x} y}{1+e^{2 x}}\) passes through \((0,0)\) and \((\ln 2, k)\), then \(k\) is

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

[JEE Main 2026, 28 Jan (Shift 2)]

If the curve satisfying the differential equation \(\frac{d y}{d x}=\frac{6-2 e^{2 x} y}{1+e^{2 x}}\) passes through \((0,0)\) and \((\ln 2, k)\), then \(k\) is

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

[JEE Main 2024, 27 Jan (Shift 2)]

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

[JEE Main 2024, 27 Jan (Shift 2)]

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

[JEE Main 2025, 23 Jan (Shift 2)]

Let a curve \(y = f(x)\) pass through the points \((0, 5) \) and \(\left({\log }_{\mathrm{e}}2,k\right).\) If the curve satisfies the differential equation \(2(3+y){e}^{2x}dx-\left(7+{e}^{2x}\right)dy=0\) then \(k\) is equal to

[JEE Main 2025, 23 Jan (Shift 1)]

Let \(x=x(y)\) be the solution of the differential equation \(y^2 d x+\left(x-\frac{1}{y}\right) d y=0\). If \(x(1)=1,\) then \(x\left(\frac{1}{2}\right)\) is:

[JEE Main 2025, 22 Jan (Shift 1)]

Let \(y=y(x)\) be the solution of the differential

equation \(\left({x}^{2}+1\right){y}^{'}-2xy=\left({x}^{4}+2{x}^{2}+1\right)\cos x\), \(y(0)=1\). Then \({\int }_{-3}^{3}y\left(x\right)dx\) is:

Let \(f:(0, \infty) \rightarrow \mathbf{R}\) be a function which is differentiable at all points of its domain and satisfies the condition \(x^2 f^{\prime}(x)=2 x f(x)+3\), with \(f(1)=4\). Then \(2 f(2)\) is equal to :

[JEE Main 2025, 24 Jan (Shift 2)]

Let \( f(x)\) be a real differentiable function such that \(f(0)=1\text{ and  }\)\(f(\mathrm{x}+\mathrm{y})=f(\mathrm{x}){f}^{'}(\mathrm{y})+{f}^{'}(\mathrm{x})f(\mathrm{y})\mathrm{for}\mathrm{all}\) \(\mathrm{x},\mathrm{y}\in R\text{. Then }\sum _{\mathrm{n}=1}^{100}{\log }_{\mathrm{e}}f\left(\mathrm{n}\right)\) is equal to:

[JEE Main 2025, 22 Jan (Shift 1)]

If \(x=f(y)\) is the solution of the differential equation

\(\left(1+y^2\right)+\left(x-2 e^{\tan ^{-1} y}\right) \frac{d y}{d x}=0, y \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\)

with \(f(0)=1\), then \(f\left(\frac{1}{\sqrt{3}}\right)\) is equal to :

[JEE Main 2025, 22 Jan (Shift 2)]

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

[JEE Main 2024, 30 Jan (Shift 1)]

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

[JEE Main 2024, 30 Jan (Shift 1)]

The solution of the differential equation \((x+1) \frac{d y}{d x}-y=e^{3 x}(x+1)^2\) is:

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

[JEE Main 2026, 2 Apr (Shift 1)]

Let \(x=x(y)\) be the solution of the differential equation \({\mathrm{y}}^{2}\mathrm{dx}+\left(\mathrm{x}-\frac{1}{\mathrm{y}}\right)\mathrm{dy}=0\). If \(x(1)=1,\) then \(x\left(\frac{1}{2}\right)\) is:

[JEE Main 2025, 22 Jan (Shift 1)]

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

[JEE Main 2026, 2 Apr (Shift 2)]

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

[JEE Main 2025, 3 Apr (Shift 2)]

Let \(y = y(x)\) be the solution of the differential equation \(\cos x{\left({\log }_{e}(\cos x)\right)}^{2}dy\)\(+\left(\sin x-3y\sin x{\log }_{e}(\cos x)\right)dx=0\), \(\mathrm{x}\in \left(0,\frac{\pi }{2}\right)\text{. If }\mathrm{y}\left(\frac{\pi }{4}\right)=\frac{-1}{{\log }_{\mathrm{e}}2}\text{, then }\mathrm{y}\left(\frac{\pi }{6}\right)\) is:

[JEE Main 2025, 29 Jan (Shift 1)]

Let \( f(x)\) be a real differentiable function such that \(f(0)=1\text{ and  }\)\(f(\mathrm{x}+\mathrm{y})=f(\mathrm{x}){f}^{'}(\mathrm{y})+{f}^{'}(\mathrm{x})f(\mathrm{y})\mathrm{for}\mathrm{all}\) \(\mathrm{x},\mathrm{y}\in R\text{. Then }\sum _{\mathrm{n}=1}^{100}{\log }_{\mathrm{e}}f(\mathrm{n})\) is equal to:

[JEE Main 2025, 22 Jan (Shift 1)]

\(\begin{aligned}&f(y) \text { is the solution of differential equation }\\&\left(1+y^2\right)+\left(x-2 \tan ^{-1} y\right) \frac{d y}{d x}=0, f(0)=1, \text { find } f\left(\frac{1}{\sqrt{3}}\right) .\end{aligned}\) (22 Jan, Shift II, Memory Based)

The population \(p(t)\) at time \(t\) of a certain mouse species satisfies the differential equation:

\(\frac{d p(t)}{d t}=0.5 p(t)-450\)

If \(p(0)=850\), then the time at which the population becomes zero is:

\(\text { If } 2 \cos x \frac{d y}{d x}=\sin 2 x-4 y \sin x \cdot y\left(\frac{\pi}{3}\right)=0 \text { find } y^{\prime}\left(\frac{\pi}{4}\right)+y\left(\frac{\pi}{4}\right)\).

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

[JEE Main 2025, 29 Jan (Shift 2)]

If \(\frac{d y}{d x}-y \log _e 2=2^{\sin x}(\cos x-1) \log _e 2\), then \(y\) is:

The population \(p(t)\) at time \(t\) of a certain mouse species satisfies the differential equation:

\(\frac{d p(t)}{d t}=0.5 p(t)-450\)

If \(p(0)=850\), then the time at which the population becomes zero is:

Let \(y=y(x)\) be the solution of the differential equation \(\frac{d y}{d x}=\left(1+x+x^2\right)\left(1-y+y^2\right), y(0)=\frac{1}{2}\) then \((2y(1)-1)\) is equal to:

[JEE Main 2026, 4 Apr (Shift 1)]

Let \(y = y(x)\) be the solution of the differential equation \(\left(xy-5{x}^{2}\sqrt{1+{x}^{2}}\right)dx+\left(1+{x}^{2}\right)dy=0\),\(y(0) = 0.\) Then \(y(\sqrt{3})\) is equal to

[JEE Main 2025, 24 Jan (Shift 1)]

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

[JEE Main 2026, 8 Apr (Shift 2)]

Let \(f:(0, \infty) \rightarrow \mathbf{R}\) be a function which is differentiable at all points of its domain and satisfies the condition \(x^2 f^{\prime}(x)=2 x f(x)+3\), with \(f(1)=4\). Then \(2 f(2)\) is equal to :

[JEE Main 2025, 24 Jan (Shift 2)]

Let \(y = y(x)\) be the solution of the differential equation \(\cos x{\left({\log }_{e}(\cos x)\right)}^{2}dy\)\(+\left(\sin x-3y\sin x{\log }_{e}(\cos x)\right)dx=0\), \(\mathrm{x}\in \left(0,\frac{\pi }{2}\right)\text{. If }\mathrm{y}\left(\frac{\pi }{4}\right)=\frac{-1}{{\log }_{\mathrm{e}}2}\text{, then }\mathrm{y}\left(\frac{\pi }{6}\right)\) is:

[JEE Main 2025, 29 Jan (Shift 1)]

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

[JEE Main 2024, 29 Jan (Shift 2)]

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

[JEE Main 2024, 29 Jan (Shift 2)]

Let y = f(x) be the solution of the differential equation y(x + 1)dx – x²dy = 0, y(1) = e. Then \(\lim _{\mathrm{x}\to {0}^{+}}f\left(x\right)\) is equal to

[JEE Main 2023, 29 Jan (Shift 1)]

Let \(y=y(x)\) be the solution of the differential equation \(\frac{d y}{d x}\) \(=2(y+2 \sin x-5) x-2 \cos x\) such that \(y(0)=7\). Then \(y(\pi)\) is equal to.

[JEE Main 2021, 27 Aug (Shift 1)]

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

[JEE Main 2021, 27 Jul (Shift 2)]

The slope of tangent at any point \((x, y)\) on a curve \(y=y(x)\) is \(\frac{x^2+y^2}{2 x y}, x>0\). If \(y(2)=0\), then a value of \(y(8)\) is

[JEE Main 2023, 10 Apr (Shift 1)]

If the solution curve of the differential equation \(\left(2x-10{y}^{3}\right)dy+ydx=0\), passes through the points \((0,1)\) and \((2,\beta )\), then \(\beta\) is a root of the equation:

[JEE Main 2021, 27 Aug (Shift 2)]

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

[JEE Main 2021, 20 Jul (Shift 1)]

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

[JEE Main 2021, 25 Jul (Shift 1)]

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

[JEE Main 2023, 8 Apr (Shift 1)]

If \(y=y(x)\) is the solution curve of the differential equation \(x^2 d y+\left(y-\frac{1}{x}\right) d x=0 ; x>0\) and \(y(1)=1\), then \(y\left(\frac{1}{2}\right)\) is equal to:

[JEE Main 2021, 1 Sep (Shift 2)]

The slope of tangent at any point \((x, y)\) on a curve \(y=y(x)\) is \(\frac{x^2+y^2}{2 x y}, x>0 x>0\). If \(y(2)=0\), then a value of \(y(8)\) is

[JEE Main 2023, 10 Apr (Shift 1)]

If \(y=y(x)\) is the solution of the differential equation \(\frac{d y}{d x}+(\tan x) y=\sin x, 0 \leq x \leq \frac{\pi}{3}\) with \(y(0)=0\), then \(y\left(\frac{\pi}{4}\right)\)

[JEE Main 2021, 16 Mar (Shift 2)]

The population \( p(t) \) at time \( t \) of a certain mouse species satisfies the differential equation \( \frac{d p(t)}{d t}=0.5 p(t)-450 \). If \( p(0)=850 \), then the time at which the population becomes zero is

[JEE Main 2021, 24 Feb (Shift 1)]

Which of the following is true for \(y(x)\) that satisfies the differential equation \(\frac{\mathrm{dy}}{\mathrm{dx}}=\mathrm{xy}-1+\mathrm{x}-\mathrm{y};\mathrm{y}\left(0\right)=0\) :

[JEE Main 2021, 17 Mar (Shift 1)]

The integrating factor of the differential equation \((x\log x)\frac{dy}{dx}+y=2\log x\), is given by

The solution of the differential equation \(\frac{\mathrm{dy}}{\mathrm{dx}}-\frac{\mathrm{y}+3\mathrm{x}}{{\log }_{\mathrm{e}}(\mathrm{y}+3\mathrm{x})}+3=0\) is: (where C is a constant of integration.)

[JEE Main 2020, 4 Sep (Shift 2)]

Let \(y=f(x)=\sin ^3\left(\frac{\pi}{3}\left(\cos \left(\frac{\pi}{3 \sqrt{2}}\left(-4 x^3+5 x^2+1\right)^{\frac{3}{2}}\right)\right)\right)\). Then, at \(x=1\),

[JEE Main 2023, 31 Jan (Shift 1)]

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

[JEE Main 2021, 18 Mar (Shift 2)]

The area enclosed by the closed curve \(C\) 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 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 \(R S\) is

[JEE Main 2023, 1 Feb (Shift 1)]

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

[JEE Main 2020, 7 Jan (Shift 1)]

Let \(y = y(x)\) be the solution of the differential equation \({\csc }^{2}\mathrm{xdy}+2\mathrm{dx}=(1+\mathrm{ycos}2\mathrm{x}){\csc }^{2}\mathrm{xdx}\), with \(\mathrm{y}\left(\frac{\pi }{4}\right)=0\). Then, the value of \((y(0) + 1)^2 \) is equal to:

[JEE Main 2021, 22 Jul (Shift 2)]

If \( f^{\prime}(x)=\tan ^{-1}(\sec x+\tan x),-\frac{\pi}{2}

[JEE Main 2020, 9 Jan (Shift 1)]

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 increased by 20% in 2 hours. If the population of bacteria is 2000 after \(\frac{\mathrm{k}}{{\log }_{\mathrm{e}}\left(\frac{6}{5}\right)}\) hours, then \({\left(\frac{\mathrm{k}}{{\log }_{\mathrm{e}}2}\right)}^{2}\) is equal to:

[JEE Main 2021, 26 Feb (Shift 1)]

Let \(\mathrm{y}=\mathrm{f}\left(\mathrm{x}\right)={\sin }^{3}\left(\frac{\pi }{3}\left(\cos \left(\frac{\pi }{3\sqrt{2}}{\left(-4{\mathrm{x}}^{3}+5{\mathrm{x}}^{2}+1\right)}^{\frac{3}{2}}\right)\right)\right)\). Then, at x = 1,

[JEE Main 2023, 31 Jan (Shift 1)]

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

[JEE Main 2023, 12 Apr (Shift 1)]

If the solution curve of the differential equation \(\left(2 x-10 y^3\right)\) \(d y+y d x=0\), passes through the points \((0,1)\) and \((2, \beta)\), then \(\beta\) is a root of the equation:

[JEE Main 2021, 27 Aug (Shift 2)]

If \(y\frac{\text{dy}}{\text{dx}} = x\left\lbrack \frac{y^{2}}{x^{2}} + \frac{\phi\left( \frac{y^{2}}{x^{2}} \right)}{\phi'\left( \frac{y^{2}}{x^{2}} \right)}\right\rbrack,x > 0,\phi > 0 \) and \(y(1) = - 1\), then \(\phi\left( \frac{y^{2}}{4} \right)\) is equal to:

[JEE Main 2021, 31 Aug (Shift 2)]

Let \(y = y(x)\) be the solution of the differential equation (3y² – 5x²)ydx + 2x (x² – y²) dy = 0 such that \(y(1) = 1\). then \(∣\left(y\left(2\right){)}^{3}-12\mathrm{y}(2)∣\right.\) is equal to:

[JEE Main 2023, 31 Jan (Shift 2)]

If \(\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{{2}^{\mathrm{x}}\mathrm{y}+{2}^{\mathrm{y}}\cdot {2}^{\mathrm{x}}}{{2}^{\mathrm{x}}+{2}^{\mathrm{x}+\mathrm{y}}{\log }_{\mathrm{e}}2},\mathrm{y}\left(0\right)=0\), then for \(y = 1\), the value of \(x\) lies in the interval:

[JEE Main 2021, 31 Aug (Shift 2)]

If \(\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{{2}^{\mathrm{x}+\mathrm{y}}-{2}^{\mathrm{x}}}{{2}^{\mathrm{y}}},\mathrm{y}(0)=1\), then \(y(1)\) is equal to:

[JEE Main 2021, 31 Aug (Shift 1)]

Let \(f\) be a differentiable function such that \(x^2 f(x)-x=4 \int_0^x t f(t) d t, f(1)=\frac{2}{3}\). Then \(18 f(3)\) is equal to

[JEE Main 2023, 10 Apr (Shift 1)]

Let x = x(y) be the solution of the differential equation 2(y + 2)log_e (y + 2)dx + (x + 4–2log_e (y + 2))dy = 0, y > –1 with x(e⁴–2) = 1. Then x(e⁹–2) is equal to

[JEE Main 2023, 15 Apr (Shift 1)]

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

[JEE Main 2021, 27 Jul (Shift 2)]

The general solution of the differential equation \(\ \left(\tan ^{-1} y-x\right) d y=\left(1+y^2\right) d x \) is

Let y = y(x) be the solution of the differential equation \(\left({\mathrm{x}}^{2}-3{\mathrm{y}}^{2}\right)\mathrm{dx}+3\mathrm{xydy}=0,\mathrm{y}(1)=1\). Then \(6{\mathrm{y}}^{2}(\mathrm{e})\) is equal to

The solution of the differential equation \((x+1) \frac{d y}{d x}-y=e^{3 x}(x+1)^2\) is

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

[JEE Main 2020, 3 Sep (Shift 1)]

Let \(y=y(x)\) be the solution of the differential equation. \(x^3 d y+(x y-1) d x=0, x>0, y\left(\frac{1}{2}\right)=3-e\). Then \(y(1)\) is equal to

[JEE Main 2023, 24 Jan (Shift 1)]

The solution of the differential equation \(\frac{\mathrm{dy}}{\mathrm{dx}}=-\left(\frac{{\mathrm{x}}^{2}+3{\mathrm{y}}^{2}}{3{\mathrm{x}}^{2}+{\mathrm{y}}^{2}}\right),\mathrm{y}(1)=0\) is

[JEE Main 2023, 30 Jan (Shift 2)]

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 increased by \(20\%\) in \(2\) hours. If the population of bacteria is \(2000\) after \(\frac{\mathrm{k}}{{\log }_{\mathrm{e}}\left(\frac{6}{5}\right)}\) hours, then \({\left(\frac{\mathrm{k}}{{\log }_{\mathrm{e}}2}\right)}^{2}\) is equal to:

[JEE Main 2021, 26 Feb (Shift 1)]

Let \( y=y(x) \) be the solution of the differential equation

\( \operatorname{cosec}{ }^{2} x d y+2 d x=(1+y \cos 2 x) \operatorname{cosec} ^{2}x d x \), with

\( y\left(\frac{\pi}{4}\right)=0 \). Then, the value of \( (y(0)+1)^{2} \) is equal

to:

[JEE Main 2021, 22 Jul (Shift 2)]

If the slope of the tangent to the curve at any point \(\ P(x, y) \) is \(\ \frac{y}{x}-\cos ^2 \frac{y}{x} \), then the equation of a curve passing through \(\ \left(1, \frac{\pi}{4}\right) \) is

Let \( y=y(x) \) be the solution curve of the differential equation \[ \frac{d y}{d x}=\frac{y}{x}\left(1+x y^{2}\left(1+\log _{e} x\right)\right), x>0, y(1)=3 \text {. } \] Then \( \frac{y^{2}(x)}{9} \) is equal to

[JEE Main 2023, 25 Jan (Shift 1)]

\(\text{ Let }y=y(x)\text{ be the solution of the differential equation }\)

\(\frac{dy}{dx}=1+x{e}^{y-x},-\sqrt{2}

[JEE Main 2021, 25 Jul (Shift 1)]

Let \(y=y(x)\) be the solution of the differential equation \({e}^{x}\sqrt{1-{y}^{2}}dx+\left(\frac{y}{x}\right)dy=0,y(1)=-1\) Then the value of \((y(3){)}^{2}\) is equal to:

[JEE Main 2021, 20 Jul (Shift 1)]

Which of the following is true for \(y(x)\) that satisfies the differential equation \(\frac{\mathrm{dy}}{\mathrm{dx}}=\mathrm{xy}-1+\mathrm{x}-\mathrm{y};\mathrm{y}(0)=0\) :

[JEE Main 2021, 17 Mar (Shift 1)]

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 equal to

If a curve \(y=f(x)\), passing through the point \((1,2)\), is the solution of the differential equation, \(2 x^2 d y=\left(2 x y+y^2\right)\) \(d x\), then \(f\left(\frac{1}{2}\right)\) is equal to:

[JEE Main 2020, 2 Sep (Shift 2)]

The number of solutions of \(\frac{d y}{d x}=\frac{y+1}{x-1}\), when \(y(1)=\mathbf{2}\) is

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

[JEE Main 2021, 18 Mar (Shift 1)]

Let \(y=y(x)\) be the solution of the differential equation \(\cos x \frac{d y}{d x}+2 y \sin x=\sin 2 x, x \in\left(0, \frac{\pi}{2}\right)\). If \(y(\frac{\pi} { 3})=0\), then \(y(\frac{\pi} { 4})\) is equal to:

[JEE Main 2020, 5 Sep (Shift 2)]

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

[JEE Main 2020, 8 Jan (Shift 1)]

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 \(3x+4y = 5\), is given by :

[JEE Main 2021, 27 Aug (Shift 2)]

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 \(y(2)\) is equal to

[JEE Main 2023, 11 Apr (Shift 2)]

Let \(y=y(x)\) be the solution curve of the differential equation \(\frac{d y}{d x}=\frac{y}{x}\left(1+x y^2\left(1+\log _e x\right)\right), x>0, y(1)=3\). Then \(\frac{y^2(x)}{9}\) is equal to:

[JEE Main 2023, 25 Jan (Shift 1)]

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

[JEE Main 2020, 7 Jan (Shift 2)]

Let \(y = y(x)\) be the solution of the differential equation \(\mathrm{xdy}=\left(\mathrm{y}+{\mathrm{x}}^{3}\mathrm{cosx}\right)\mathrm{dx}\) with \(y(\pi) = 0\), then \(\mathrm{y}\left(\frac{\pi }{2}\right)\)is equal to:

[JEE Main 2021, 25 Jul (Shift 2)]

\(\text{ Let }y=y(x)\text{ be solution of the differential equation }{\log }_{e}\left(\frac{dy}{dx}\right)=3x+4y\text{, with }y(0)=0\text{. }\)

\(\text{ If }y\left(-\frac{2}{3}{\log }_{e}2\right)=\alpha {\log }_{e}2\text{, then the value of }\alpha \text{ is equal to: }\)

[JEE Main 2021, 27 Jul (Shift 1)]

Let \(y=y(x)\) be the solution of the differential equation

\(\cos x(3 \sin x+\cos x+3) dy=(1+y \sin x(3 \sin x+\cos x+3))\) \(d x, 0 \leq x \leq \frac{\pi}{2}, y(0)=0\).

Then \(y\left(\frac{\pi}{3}\right)\) is equal to:

[JEE Main 2021, 17 Mar (Shift 2)]

If the curve \(y=y(x)\) is the solution of the differential equation \(2\left({x}^{2}+{x}^{5/4}\right)dy-y\left(x+{x}^{1/4}\right)dx=2{x}^{9/4}dx,x>0\) which passes through the point \(\left(1,1-\frac{4}{3}{\log }_{e}2\right)\), then the value of \(y(16)\) is equal to:

[JEE Main 2021, 17 Mar (Shift 2)]

If \(y=y(x)\) is the solution curve of the differential equation \(x^2 d y+\left(y-\frac{1}{x}\right) d x=0 ; x>0\) and \(y(1)=1\), then \(y\left(\frac{1}{2}\right)\) is equal to:

Let \(y(x)\) be the solution of the differential equation \(2{x}^{2}dy+\left({e}^{y}-2x\right)dx=0,x>0\). If \(y(e)=1\), then \(y(1)\) is equal to:

[JEE Main 2021, 26 Aug (Shift 2)]

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

[JEE Main 2023, 15 Apr (Shift 1)]

Let \(y=y(x)\) be in the solution of the differential equation \(\cos x(3\sin x+\cos x+3)dy=(1+y\sin x(3\sin x+\cos x+3))dx,0\leq x\leq \frac{\pi }{2},y(0)=0\).

Then \(y\left(\frac{\pi }{3}\right)\) is equal to:

[JEE Main 2021, 17 Mar (Shift 2)]

Let \(y(x)\) be the solution of the differential equation \(2 x^2 d y\) \(+\left(e^y-2 x\right) d x=0, x>0\). If \(y(e)=1\), then \(y(1)\) is equal to:

[JEE Main 2021, 26 Aug (Shift 2)]

Let \(f:[1, \infty) \rightarrow \mathbb{R}\) be a differentiable function such that \(f(1)=\frac{1}{3}\) and \(3 \int_1^x f(t) d t=x f(x)-\frac{x^3}{3}, x \in[1, \infty)\). Let \(e\) denote the base of the natural logarithm. Then the value of \(f(e)\) is

[JEE Advanced 2023]

If \({\mathrm{x}}^{3}\mathrm{dy}+\mathrm{xydx}={\mathrm{x}}^{2}\mathrm{dy}+2\mathrm{ydx};\mathrm{y}(2)=\mathrm{e}\) and \(x > 1,\) then \(y(4)\) is equal to:

[JEE Main 2020, 3 Sep (Shift 2)]

Let \(\alpha x=\exp \left(x^\beta y^\gamma\right)\) be the solution of the differential equation \(2 x^2 y d y-\left(1-x y^2\right) d x=0, x>0, y(2)=\sqrt{\log _e 2}\). Then \(\alpha+\beta-\gamma\) equals:

[JEE Main 2023, 1 Feb (Shift 2)]

Let \( \mathrm{f} \) be a twice differentiable function defined on \( R \)

such that \( f(0)=1, f^{\prime}(0)=2 \) and \( f^{\prime}(\mathrm{x}) \neq 0 \) for all

\( x \in R \). If \( \left|\begin{array}{ll}f(x) & f^{\prime}(x) \\ f^{\prime}(x) & f^{\prime \prime}(x)\end{array}\right|=0 \), for all \( x \in R \), then the

value of f(1) lies in the interval

[JEE Main 2021, 24 Feb (Shift 2)]

If \(\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{{2}^{\mathrm{x}}\mathrm{y}+{2}^{\mathrm{y}}\cdot {2}^{\mathrm{x}}}{{2}^{\mathrm{x}}+{2}^{\mathrm{x}+\mathrm{y}}{\log }_{\mathrm{e}}2},\mathrm{y}(0)=0\), then for \(y = 1\), the value of \(x\) lies in the interval:

[JEE Main 2021, 31 Aug (Shift 2)]

If \(y=y(x)\) is the solution curve of the differential equation \(\frac{dy}{dx}+y\tan x=x\sec x,0\leq x\leq \frac{\pi }{3}\), y (0) = 1, then \(y\left(\frac{\pi }{6}\right)\) is equal to

If \(\frac{d y}{d x}=\frac{x y}{x^2+y^2} ; y(1)=1\); then a value of \(x\) satisfying \(y(x)=e\) is:

[JEE Main 2020, 9 Jan (Shift 2)]

The general solution of the differential equation \(\sqrt{1+{\mathrm{x}}^{2}+{\mathrm{y}}^{2}+{\mathrm{x}}^{2}{\mathrm{y}}^{2}}+\mathrm{xy}\frac{\mathrm{dy}}{\mathrm{dx}}=0\) is: (where C is a constant of integration)

[JEE Main 2020, 6 Sep (Shift 1)]

Let \(y=y(t)\) be a solution of the differential equation \(\frac{\mathrm{dy}}{\mathrm{dt}}+\alpha y=\gamma \mathrm{e}^{-\beta t}\) Where, \(\alpha>0, \beta>0\) and \(\gamma>0\). Then \(\lim _{t \rightarrow \infty} y(t)\)

[JEE Main 2023, 25 Jan (Shift 2)]

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 \(f(x)+x f^{\prime}(x)=x^2\). Then:

[JEE Main 2021, 27 Aug (Shift 1)]

If \(y\frac{\text{dy}}{\text{dx}} = x\left\lbrack \frac{y^{2}}{x^{2}} + \frac{\phi\left( \frac{y^{2}}{x^{2}} \right)}{\phi'\left( \frac{y^{2}}{x^{2}} \right)},x > 0,\phi > 0 \right\rbrack\) and \(y(1) = - 1\), then \(\phi\left( \frac{y^{2}}{4} \right)\) is equal to:

[JEE Main 2021, 31 Aug (Shift 2)]

Let \(y=y(x)\) be the solution of the differential equation \(\mathrm{xtan}\left(\frac{\mathrm{y}}{\mathrm{x}}\right)\mathrm{dy}=\left(\mathrm{ytan}\left(\frac{\mathrm{y}}{\mathrm{x}}\right)-\mathrm{x}\right)\mathrm{dx},-1\leq \mathrm{x}\leq 1,\mathrm{y}\left(\frac{1}{2}\right)=\frac{\pi }{6}\) Then the area of the region bounded by the curves \(x=0,x=\frac{1}{\sqrt{2}}\) and \(y=y(x)\) in the upper half plane is:

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

[JEE Main 2020, 2 Sep (Shift 1)]

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

[JEE Main 2021, 18 Mar (Shift 2)]

Let \(y=y(x)\) be the solution of the differential equation \(\frac{dy}{dx}=1+x{e}^{y-x},-\sqrt{2}

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

[JEE Main 2023, 15 Apr (Shift 1)]

Let \(f(x)=\int_0^x e^t f(t) d t+e^x\) be a differentiable function for all \(x \in R\). Then \(f(x)\) equals:

[JEE Main 2021, 26 Feb (Shift 2)]

Let \(y = y(x)\) be the solution of the differential equation \((3{y}^{2}–5{x}^{2})ydx+2x({x}^{2}–{y}^{2})dy=0\) such that \(y(1) = 1\). then \(\left|(y(2))^3-12 y(2)\right|\) is equal to:

[JEE Main 2023, 31 Jan (Shift 2)]

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

Let \(y=y(x)\) be the solution of the differential equation

\(x\tan \left(\frac{y}{x}\right)dy=\left(y\tan \left(\frac{y}{x}\right)-x\right)dx,-1\leq x\leq 1,y\left(\frac{1}{2}\right)=\frac{\pi }{6}\)

Then the area of the region bounded by the curves \(x=0,x=\frac{1}{\sqrt{2}}\) and \(y=y(x)\) in the upper half plane is :

[JEE Main 2021, 20 Jul (Shift 1)]

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

[JEE Main 2021, 20 Jul (Shift 2)]

The number of solutions of\(\text{ }\frac{dy}{dx}=\frac{y+1}{x−1}\), when \(y(1)=2\) is

If a curve \( y=f(x) \) passes through the point \( (1,2) \) and satisfies \( x \frac{d y}{d x}+y=b x^{4} \), then for what value of \( b \), \( \int_{1}^{2} f(x) d x=\frac{62}{5} ? \)

[JEE Main 2021, 24 Feb (Shift 2)]

Let \(f(x)={\int }_{0}^{x}{e}^{t}f(t)dt+{e}^{x}\) be a differentiable function for all \(x\in R\). Then \(f(x)\) equals:

[JEE Main 2021, 26 Feb (Shift 2)]

The solution of the differential equation \(\frac{dy}{dx}=−\left(\frac{{x}^{2}+3{y}^{2}}{3{x}^{2}+{y}^{2}}\right),\text{ }y\left(1\right)=0\) is

A particle is moving in the \(x y\)-plane along a curve \(C\) passing through the point \((3,3)\). The tangent to the curve \(C\) at the point \(P\) meets the \(x\)-axis at \(Q\). If the \(y\)-axis bisects the segment \(P Q\), then \(C\) is a parabola with

[JEE Main 2022, 24 June (Shift 2)]

If \( y=\left(\frac{2}{\pi} x-1\right) \operatorname{cosec} x \) is the solution of the differential equation, \( \frac{d y}{d x}+p(x) y=\frac{2}{\pi} \operatorname{cosec} x, 0

[JEE Main 2020, 6 Sep (Shift 2)]

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 equal to

[JEE Main 2023, 1 Feb (Shift 1)]

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 \(y_1(0)=0, y_2(0)=1\) respectively. Then the curves \(y=y_1\) \((x)\) and \(y=y_2(x)\) intersect at

[JEE Main 2023, 13 Apr (Shift 1)]

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

[JEE Main 2020, 3 Sep (Shift 1)]

The solution of the differential equation \(\frac{\mathrm{dy}}{\mathrm{dx}}=-\left(\frac{{\mathrm{x}}^{2}+3{\mathrm{y}}^{2}}{3{\mathrm{x}}^{2}+{\mathrm{y}}^{2}}\right),\mathrm{y}\left(1\right)=0\) is

[JEE Main 2023, 30 Jan (Shift 2)]

Let \(y=y(x)\) be the solution of the differential equation \(\left(x^2-3 y^2\right) d x+3 x y d y=0, y(1)=1\).
Then \(6 y^2(\mathrm{e})\) is equal to

[JEE Main 2023, 24 Jan (Shift 2)]

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

[JEE Main 2020, 8 Jan (Shift 2)]

Let \(y=y(x)\) be the solution of the differential equation \(xdy=\left(y+{x}^{3}\cos x\right)dx\) with \(y(\pi )=0\), then \(y\left(\frac{\pi }{2}\right)\) is equal to :

[JEE Main 2021, 25 Jul (Shift 2)]

Let the solution curve \(y=y(x)\) of the differential equation \(\frac{d y}{d x}-\frac{3 x^5 \tan ^{-1}\left(x^3\right)}{\left(1+x^6\right)^{\frac{3}{2}}} y=2 x \cdot \exp \left(\frac{x^3-\tan ^{-1} x^3}{\sqrt{1+x^6}}\right)\) pass through the origin. Then \(y(1)\) is equal to:

[JEE Main 2023, 30 Jan (Shift 1)]

Let \( y=y(x) \) satisfies the equation \( \frac{d y}{d x}-|A|=0 \), 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 \( y(\pi)=\pi+2 \), then

the value of \( y\left(\frac{\pi}{2}\right) \) is:

[JEE Main 2021, 20 Jul (Shift 2)]

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

[JEE Main 2021, 20 Jul (Shift 2)]

Let \(y=y(x)\) be the solution of the differential equation \(\mathrm{xtan}\left(\frac{\mathrm{y}}{\mathrm{x}}\right)\mathrm{dy}=\left(\mathrm{ytan}\left(\frac{\mathrm{y}}{\mathrm{x}}\right)-\mathrm{x}\right)\mathrm{dx},-1\leq \mathrm{x}\leq 1,\mathrm{y}\left(\frac{1}{2}\right)=\frac{\pi }{6}\). Then the area of the region bounded by the curves \(x=0,x=\frac{1}{\sqrt{2}}\) and \(y=y(x)\) in the upper half plane is:

If \(y=y(x)\) is the solution curve of the differential equation \({x}^{2}dy+\left(y-\frac{1}{x}\right)dx=0;x>0\), and \(y(1)=\) 1 , then \(y\left(\frac{1}{2}\right)\) is equal to:

[JEE Main 2021, 1 Sep (Shift 2)]

Let y = y(x) be the solution of the differential equation (3y² – 5x²)y dx + 2x(x² – y²)dy = 0 such that y(1) = 1. Then |y(2))³ – 12y(2)| is equal to

[JEE Main 2023, 31 Jan (Shift 2)]

Let, y =f (x) be the solution of the differential equation \(y(x+1)dx−{x}^{2}dy=0,y(1)=e.\) Then \(\lim _{x\to {0}^{+}}f(x)\)is equal to

[JEE Main 2023, 29 Jan (Shift 1)]

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

[JEE Main 2021, 16 Mar (Shift 1)]

Let y=y(x) 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(x) at y=2 and the line x=2 intersects the curve y=y(x) at \(y=\alpha\), then a value of \(\alpha\) is

[JEE Main 2023, 11 Apr (Shift 1)]

The solution of the differential equation \(\frac{\mathrm{dy}}{\mathrm{dx}}-\frac{\mathrm{y}+3\mathrm{x}}{{\log }_{\mathrm{e}}(\mathrm{y}+3\mathrm{x})}+3=0\) is: (where \(C\) is a constant of integration.)

[JEE Main 2020, 4 Sep (Shift 2)]

Let \( y=y(x) \) be the Solutions of the differential equation, \( x y' -y=x^{2}(x \cos x+\sin x), x>0 \). If \( y(\pi)=\pi \), then. \( y^{\prime \prime}\left(\frac{\pi}{2}\right)+y\left(\frac{\pi}{2}\right) \) is equal to :

[JEE Main 2020, 4 Sep (Shift 1)]

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 \((\alpha, 2)\) then \(\alpha^\alpha\) is equal to

[JEE Main 2023, 6 Apr (Shift 2)]

Let \(y = y(x)\) be solution of the differential equation \({\log }_{\mathrm{e}}\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)=3\mathrm{x}+4\mathrm{y}\) with \(y(0) = 0\). If\(\mathrm{y}\left(-\frac{2}{3}{\log }_{\mathrm{e}}2\right)={\mathrm{αlog}}_{\mathrm{e}}2\), then the value of \(α\) is equal to:

[JEE Main 2021, 27 Jul (Shift 1)]

If the curve \(y=y(x)\) is the solution of the differential equation \(2\left(x^2+x^{5 / 4}\right) d y-y\left(x+x^{1 / 4}\right) d x=2 x^{9 / 4} d x, x>0\)

which passes through the point \(\left(1,1-\frac{4}{3} \log _e 2\right)\), then the value of \(y(16)\) is equal to:

[JEE Main 2021, 17 Mar (Shift 2)]

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

[JEE Main 2021, 18 Mar (Shift 2)]

Let \(y=f(x)\) be the solution of the differential equation \(y(x+1)dx–{x}^{2}dy=0,y(1)=e.\) Then \(\lim _{\mathrm{x}\to {0}^{+}}f\left(x\right)\) is equal to

[JEE Main 2023, 29 Jan (Shift 1)]

The solutions of \((x+y+1) d y=d x\) are

Let y = y(x) be a solution curve of the differential equation, \(\left(1-{\mathrm{x}}^{2}{\mathrm{y}}^{2}\right)\mathrm{dx}=\mathrm{ydx}+\mathrm{xdy}\). 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 = α, then a value of α is

[JEE Main 2023, 11 Apr (Shift 1)]

Which of the following is true for \(y(x)\) that satisfies the differential equation \(\frac{dy}{dx}=xy-1+x-y;y(0)=0\) :

[JEE Main 2021, 17 Mar (Shift 1)]

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

[JEE Main 2021, 31 Aug (Shift 1)]

Let \(y=y(x)\) be a solution curve of the differential equation \((y+1) \tan ^2 x d x+\tan x d y+y d x=0, x \in\left(0, \frac{\pi}{2}\right)\). If \(\lim _{x \rightarrow 0^{+}} x y(x)\) \(=1\), then the value of \(y\left(\frac{\pi}{4}\right)\) is:

[JEE Main 2021, 26 Aug (Shift 1)]

Let \(y=y(x)\) be the solution of the differential equation \(x \log _e x \frac{d y}{d x}+y=x^2 \log _e x,(x>1)\). If \(y(2)=2\), then \(y(e)\) is equal to

[JEE Main 2023, 29 Jan (Shift 2)]

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

[JEE Main 2021, 20 Jul (Shift 1)]

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

[JEE Main 2023, 12 Apr (Shift 1)]

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) form the line \(3x+4y = 5\), is given by :

[JEE Main 2021, 27 Aug (Shift 2)]

If \(\frac{dy}{dx}=\frac{{2}^{x}y+{2}^{y}\cdot {2}^{x}}{{2}^{x}+{2}^{x+y}{\log }_{c}2},y(0)=0\), then for \(y=1\), the value of \( x \) lies in the interval:

[JEE Main 2021, 31 Aug (Shift 2)]

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

[JEE Main 2021, 18 Mar (Shift 1)]

The slope of the tangent to a curve \(C:y = y(x)\) at any point \(( x,y)\) on it is \(\frac{2e^{2x} - 6e^{- x} + 9}{2 + 9e^{- 2x}}\). If \(C\) passes through the points \(\left( 0,\frac{1}{2} + \frac{\pi}{2 \sqrt{2}} \right)\) and \(\left( \alpha,\frac{1}{2}e^{2\alpha} \right)\) then \(e^{\alpha}\) is equal to :

[JEE Main 2022, 25 Jul (Shift 1)]