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

If \( y=y(x),\ y\in\left[0,\dfrac{\pi}{2}\right] \) is the solution of the differential equation \( \sec y\,\dfrac{dy}{dx}-\sin(x+y)-\sin(x-y)=0 \), with \( y(0)=0 \), then \( 5y'\!\left(\dfrac{\pi}{2}\right) \) is equal to ________.

Let \( y = y(x) \) be the solution of the differential equation \( \text{cosec}^{2}x \, dy + 2 \, dx = (1 + y \cos 2x)\text{cosec}^{2}x \, dx \), with \( y\!\left(\dfrac{\pi}{4}\right)=0 \). Then, the value of \( (y(0)+1)^{2} \) is equal to:

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

Let \( y = y(x) \) be the solution of the differential equation
\( \left({(x+2)}\, e^{\left(\dfrac{y+1}{x+2}\right)} + (y+1) \right) dx = (x+2)\,dy, \; y(1)=1. \)
If the domain of \( y = y(x) \) is an open interval \( (\alpha, \beta) \), then \( | \alpha + \beta | \) is equal to _______.

Let \( y=y(x) \) be solution of the following differential equation \( e^{y}\dfrac{dy}{dx}-2e^{y}\sin x+\sin x \cos^{2} x=0 , \; y\!\left(\dfrac{\pi}{2}\right)=0 \).
If \( y(0)=\log_{e}\!\left(\alpha+\beta e^{-2}\right) \), then \( 4(\alpha+\beta) \) is equal to____________ .

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

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

If \( y^{1/4} + y^{-1/4} = 2x \), and \( (x^2 - 1)\dfrac{d^2 y}{dx^2} + \alpha x \dfrac{dy}{dx} + \beta y = 0 \), then \( |\,\alpha - \beta\,| \) is equal to \( \_\_\_\_\_\_\_\_\_\_ \).

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

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

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

Let \( y=y(x) \) satisfy the equation \( \dfrac{dy}{dx}-|A|=0 \), for all \( x>0 \), where .If \( y(\pi)=\pi+2 \), then the value of \( y\!\left(\dfrac{\pi}{2}\right) \) is:

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

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:

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

Let \( y = y(x) \) be a solution curve of the differential equation \( (y+1)\tan^2 x \, dx + \tan x \, dy + y \, dx = 0, \, x \in \left(0, \frac{\pi}{2}\right) \). If \( \displaystyle \lim_{x \to 0^+} x y(x) = 1 \), then the value of \( y\!\left(\frac{\pi}{4}\right) \) is:

Let a curve \( y=y(x) \) be given by the solution of the differential equation \( \cos\!\left(\dfrac{1}{2}\cos^{-1}\!\left(e^{-x}\right)\right)dx=\sqrt{e^{2x}-1}\,dy \). If it intersects y-axis at \( y=-1 \), and the intersection point of the curve with x-axis is \( (\alpha,0) \), then \( e^{\alpha} \) is equal to ______.

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

If \( [x] \) denotes the greatest integer less than or equal to \( x \), then the value of the integral \( \displaystyle\int\limits_{-\pi/2}^{\pi/2} \big[[x]-\sin x\big]\,dx \) is equal to:

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

Let \( C_1 \) be the curve obtained by the solution of differential equation \( 2xy \, \dfrac{dy}{dx} = y^2 - x^2, \, x>0 \). Let the curve \( C_2 \) be the solution of \( \dfrac{2xy}{x^2-y^2} = \dfrac{dy}{dx} \). If both the curves pass through \( (1,1) \), then the area enclosed by the curves \( C_1 \) and \( C_2 \) is equal to:

Let a curve \( y = f(x) \) pass through the point \( (2, (\log_e 2)^2) \) and have slope \( \dfrac{2y}{x \log_e x} \) for all positive real value of \( x \). Then the value of \( f(e) \) is equal to ________.

Let \( y=y(x) \) be the solution of the differentiable equation \( \cos x\, (3\sin x+\cos x+3)\,dy=(1+y\sin x\,(3\sin x+\cos x+3))\,dx, \; 0\le x\le \dfrac{\pi}{2}, \; y(0)=0 \). Then, \( y\!\left(\dfrac{\pi}{3}\right) \) is equal to:

Let \( g(t)=\displaystyle\int\limits_{-\pi/2}^{\pi/2} \cos\!\left(\frac{\pi}{4}\, t+ f(x)\right)\,dx \), where \( f(x)=\log_{e}\!\left(x+\sqrt{x^{2}+1}\right),\; x\in \mathbb{R} \). Then which one of the following is correct?

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

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

Let \( y=y(x) \) be the solution of the differential equal to \( dy=e^{\alpha x + y} \, dx:\ \alpha \in \mathbb{N} \). If \( y(\log_{e} 2)=\log_{e} 2 \) and \( y(0)=\log_{e}\!\left(\dfrac{1}{2}\right) \), then the value of \( \alpha \) is equal to ____________.

If \( y = y(x) \) is the solution of the differential equation \( \dfrac{dy}{dx} + (\tan x) \, y = \sin x, \, 0 \le x \le \dfrac{\pi}{3} \) with \( y(0)=0 \), then \( y\!\left(\dfrac{\pi}{4}\right) \) equal to:

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

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 \( f(x)+x f'(x)=x^2 \). Then:

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JEE Main 2021 25th July Evening Shift

Let a curve \( y = f(x) \) pass through the point \( (2, (\log_e 2)^2) \) and have slope \( \dfrac{2y}{x \log_e x} \) for all positive real value of \( x \). Then the value of \( f(e) \) is equal to ________.