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Let \( f:[1,\infty)\to \ [2,\infty) \) be a differentiable function. If \( 10\int_{1}^{x} f(t)\,dt=5x f(x)-x^{5}-9 \) for all \( x\ge 1 \), then the value of \( f(3) \) is:

Let \( y = y_1(x) \) and \( y = y_2(x) \) be the solution curves of the differential equation \( \dfrac{dy}{dx} = y + 7 \) with initial conditions \( y_1(0) = 0 \) and \( y_2(0) = 1 \) respectively. Then the curves \( y = y_1(x) \) and \( y = y_2(x) \) intersect at.

Let \( y=y(x) \) be the solutions of the differential equation \( (\tan x)^{\tfrac{1}{2}}\,dy=\left(\sec^3 x-(\tan x)^{\tfrac{3}{2}}\,y\right)dx \),\(0 < x < \frac{\pi}{2},\)\(y\left(\frac{\pi}{4}\right) = \frac{6\sqrt{2}}{5}.\) If \(y\left(\frac{\pi}{3}\right) = \frac{4}{5}\alpha,\) \(\text{then } \alpha^4\) equals_______________

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

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

Let \( y=y_{1}(x) \) and \( y=y_{2}(x) \) be two distinct solutions of the differential equation \( \dfrac{dy}{dx}=x+y \) with \( y_{1}(0)=0 \) and \( y_{2}(0)=1 \) respectively. Then, the number of points of intersection of \( y=y_{1}(x) \) and \( y=y_{2}(x) \) is:

Let \( \alpha \) be a non-zero real number. Suppose \( f:\mathbb{R} \rightarrow \mathbb{R} \) is a differentiable function such that \( f(0) = 2 \) and \(\displaystyle \lim_{x \to -\infty} f(x) = 1 \). If \( f'(x) = \alpha f(x) + 3 \), for all \( x \in \mathbb{R} \), then \( f(-\log_{e} 2) \) is equal to ________.

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

If \( y = y(x) \) is the solution of the differential equation \( x \dfrac{dy}{dx} + 2y = x e^{x} , \, y(1) = 0 \) then the local maximum value of the function \( z(x) = x^{2} y(x) - e^{x} , \, x \in \mathbb{R} \) is:

Let \( y = y(x) \) be the solution of the differential equation \(\displaystyle \frac{dy}{dx} + \frac{5}{x(x^5 + 1)}y = \frac{(x^5 + 1)^2}{x^7} , \; x>0 \). If \( y(1) = 2 \), then \( y(2) \) is equal to:

Let \( y = y(x) \) be the solution of the differential equation \( \frac{dy}{dx} + \frac{\sqrt{2}\,y}{2\cos^4 x - \cos 2x} = x e^{\tan^{-1}(\sqrt{2}\cot 2x)} \), \( 0 < x < \frac{\pi}{2} \) with \( y\!\left(\frac{\pi}{4}\right) = \frac{\pi^2}{32} \). If \( y\!\left(\frac{\pi}{3}\right) = \frac{\pi^2}{18} e^{-\tan^{-1}(\alpha)} \), then the value of \( 3\alpha^2 \) is equal to ________.

Let a curve \( y=f(x), \, x \in (0,\infty) \) pass through the points \( P\!\left(1,\dfrac{3}{2}\right) \) and \( Q\!\left(a,\dfrac{1}{2}\right) \). If the tangent at any point \( R(b,f(b)) \) to the given curve cuts the \( y \)-axis at the point \( S(0,c) \) such that \( bc=3 \), then \( (PQ)^2 \) is equal to ________ .

If \( \dfrac{dy}{dx} + 2y \tan x = \sin x, \, 0 < x < \dfrac{\pi}{2} \) and \( y\!\left(\dfrac{\pi}{3}\right) = 0 \), then the maximum value of \( y(x) \) is:

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

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 \( y = y(x),\; x>1 \), be the solution of the differential equation \( (x-1)\frac{dy}{dx} + 2xy = \frac{1}{x-1} \) with \( y(2) = \frac{1+e^{4}}{2e^{4}} \). If \( y(3) = \frac{e^{\alpha}+1}{\beta e^{\alpha}} \), then the value of \( \alpha + \beta \) is equal to ______.

Let the solution curve \( y=f(x) \) of the differential equation \( \dfrac{dy}{dx}+\dfrac{xy}{x^2-1}=\dfrac{x^4+2x}{\sqrt{1-x^2}}, \, x \in (-1,1) \) pass through the origin. Then \( \displaystyle \int\limits_{-\dfrac{\sqrt{3}}{2}}^{\dfrac{\sqrt{3}}{2}} f(x)\,dx \) is equal to:

If the solution of the differential equation \( \dfrac{dy}{dx} + e^{x}(x^{2} - 2)y = (x^{2} - 2x)(x^{2} - 2)e^{2x} \) satisfies \( y(0) = 0 \), then the value of \( y(2) \) is ________.

The temperature \( T(t) \) of a body a time \( t = 0 \) is \( 160^\circ F \) and it decreases continuously as per the differential equation \( \dfrac{dT}{dt} = -K (T - 80) \), where \( K \) is a positive constant. If \( T(15) = 120^\circ F \), then \( T(45) \) is equal to:

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 \( y(-1)=-\dfrac{\pi}{4} \), then \( y(0) \) is equal to:

Let \( y = y(x) \) be the solution of the differential equation \( 2\cos x \dfrac{dy}{dx} = \sin 2x - 4y \sin x, \; x \in \left(0,\dfrac{\pi}{2}\right). \) If \( y\!\left(\dfrac{\pi}{3}\right)=0 \), then \( y'\!\left(\dfrac{\pi}{4}\right) + y\!\left(\dfrac{\pi}{4}\right) \) is equal to \( \underline{\phantom{xxxx}} \).

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:

Let \( y = y(x) \) be the solution of the differential equation \( \frac{dy}{dx} = \frac{(\tan x) + y}{\sin x(\sec x - \sin x \tan x)} \), \( x \in \left(0,\frac{\pi}{2}\right) \) satisfying the condition \( y\!\left(\frac{\pi}{4}\right)=2 \). Then, \( y\!\left(\frac{\pi}{3}\right) \) is:

If \( y = y(x), \ x \in (0,\pi/2) \) be the solution curve of the differential equation \( (\sin^2 2x)\dfrac{dy}{dx} + (8\sin^2 2x + 2\sin 4x)y = 2e^{-4x}(2\sin 2x + \cos 2x), \) with \( y(\pi/4)=e^{-\pi} \), then \( y(\pi/6) \) is equal to:

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 ________.

The solution of the differential equation \( (x^{2}+y^{2})dx-5xy\,dy=0, \; y(1)=0 \), is:

Let $S = (0, 2\pi) - \left\{ \frac{\pi}{2}, \frac{3\pi}{4}, \frac{3\pi}{2}, \frac{7\pi}{4} \right\}$. Let $y = y(x), x \in S$, be the solution curve of the differential equation

$\frac{dy}{dx} = \frac{1}{1 + \sin 2x}, y\left(\frac{\pi}{4}\right) = \frac{1}{2}.$

If the sum of abscissas of all the points of intersection of the curve $y = y(x)$ with the curve ($y = \sqrt{2} \sin x$) is $\frac{k\pi}{12}$, then $k$ is equal to ___________.

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

If the solution of the differential equation \( (2x + 3y - 2)\,dx + (4x + 6y - 7)\,dy = 0 \), \( y(0) = 3 \), is \( \alpha x + \beta y + 3 \log_{e} |2x + 3y - \gamma| = 6 \), then \( \alpha + 2\beta + 3\gamma \) 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:

If the solution curve of the differential equation \( \big((\tan^{-1}y)-x\big)\,dy=(1+y^{2})\,dx \) passes through the point \( (1,0) \), then the abscissa of the point on the curve whose ordinate is \( \tan(1) \), is

Let \( y = y(x) \) be the solution of the differential equation \( x(1-x^{2})\dfrac{dy}{dx} + \left(3x^{2}y - y - 4x^{3}\right) = 0 \), \( x > 1 \), with \( y(2) = -2 \). Then \( y(3) \) is equal to

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

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

For the function \( f(x) = e^{\sin|x|} - |x|, \; x \in \mathbb{R} \), consider the following statements:
Statement-I: \( f \) is differentiable for all \( x \in \mathbb{R} \).
Statement-II: \( f \) is increasing in \( \left(-\pi, \frac{\pi}{2}\right) \).
In the light of the above statements, choose the correct answer from the options given below:

Let \( Y=Y(x) \) be a curve lying in the first quadrant such that the area enclosed by the line \( Y-y=Y'(x)(X-x) \) and the co-ordinate axes, where \( (x,y) \) is any point on the curve, is always \( \dfrac{-y^2}{2Y'(x)}+1 ,\)\( Y'(x)\neq 0) \). If \( Y(1)=1 \), then \( 12Y(2) \) equals _______.

Let \( f \) be a twice differentiable non-negative function such that \( (f(x))^{2} = 25 + \int_{0}^{x}\left((f(t))^{2} + (f'(t))^{2}\right)\,dt \). Then the mean of \( f(\log_{e}(1)),\ f(\log_{e}(2)),\ \ldots,\ f(\log_{e}(625)) \) is equal to ______.

Let \( y = y(x) \) be a solution curve of the differential equation, \( (1 - x^2y^2)\,dx = y\,dx + x\,dy \). 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:

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

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

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

Let \( y = y(x) \) be the solution of the differential equation \( \sec x \, \frac{dy}{dx} - 2y = 2 + 3\sin x, \; x \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \) and \( y(0) = -\frac{7}{4} \). Then \( y\!\left(\frac{\pi}{6}\right) \) is equal to:

The solution curve of the differential equation \( y \frac{dx}{dy} = x(\log_{e} x - \log_{e} y + 1), \, x>0, y>0 \) passing through the point \( (e, 1) \) is:

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 \( \cos x(\log_e(\cos x))^2\,dy+(\sin x-3y\sin x\log_e(\cos x))\,dx=0 \), \( x\in\left(0,\frac{\pi}{2}\right) \). If \( y\left(\frac{\pi}{4}\right)=-\frac{1}{\log_e 2} \), then \( y\left(\frac{\pi}{6}\right) \) is equal to:

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

Let \( x = x(y) \) be the solution of the differential equation \( 2y^2 \frac{dx}{dy} - 2xy + x^2 = 0, \; y > 1, x(e) = e \). Then \( x(e^2) \) 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 _______.

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

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

Let \( y=y(x) \) be the solution curve of the differential equation \( \sin(2x^{2})\log_{e}(\tan x^{2})\,dy+\left(4xy-4\sqrt{2}\,x\sin\!\left(x^{2}-\dfrac{\pi}{4}\right)\right)dx=0,\; 0<x<\sqrt{\dfrac{\pi}{2}}\), which passes through the point \(\left(\sqrt{\dfrac{\pi}{6}},1\right)\). Then \(\Big|y\left(\sqrt{\dfrac{\pi}{3}}\right)\Big|\) is equal to_______.

If the solution \( y(x) \) of the given differential equation \( (e^{y}+1)\cos x\,dx+e^{y}\sin x\,dy=0 \) passes through the point \( \left(\frac{\pi}{2},0\right) \), then the value of \( e^{\,y(\frac{\pi}{6})} \) is equal to ______.

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

If \( y = y(x) \) satisfies the differential equation \( 16\left(\sqrt{{x}+{9}+\sqrt{x}}\right)\!\left(4+\sqrt{9+\sqrt{x}}\right)\cos y\,dy = (1+2\sin y)\,dx,\ x>0 \) and \( y(256)=\frac{\pi}{2},\ y(49)=\alpha \), then \( 2\sin\alpha \) 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 \( y = f(x) \) be the solution of the differential equation \( y(x+1)dx - x^2 dy = 0 \), \( y(1) = e \). Then \( \displaystyle \lim_{x \to 0^+} f(x) \) is equal to:

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

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:

Let \( y=y(x) \) be the solution of the differential equation \( x^{3}dy+(xy-1)dx=0,\, x>0,\, y\!\left(\dfrac{1}{2}\right)=3-e \). Then \( y(1) \) 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 the solution curve, of the differential equation \(\displaystyle \frac{dy}{dx}=\frac{x+y-2}{x-y} \) passing through the point \( (2,1) \) is
\(\displaystyle \tan^{-1}\!\left(\frac{y-1}{x-1}\right)-\frac{1}{\beta}\log_{e}\!\left(a+\left(\frac{y-1}{x-1}\right)^{2}\right)=\log_{e}|x-1|, \) then \( 5\beta+\alpha \) is equal to:

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

Let \( x = x(t) \) and \( y = y(t) \) be solutions of the differential equations \( \frac{dx}{dt} + a x = 0 \) and \( \frac{dy}{dt} + b y = 0 \) respectively, \( a, b \in \mathbb{R} \). Given that \( x(0) = 2 \); \( y(0) = 1 \) and \( 3y(1) = 2x(1) \). The value of \( t \), for which \( x(t) = y(t) \), is:

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

Let $y = y(x)$ be the solution of the differential equation $\left(1 - x^2\right)dy = \left[ xy + \left(x^3 + 2\right)\sqrt{3\left(1 - x^2\right)} \right] dx$, $-1 < x < 1, y(0) = 0$. If $y\left(\frac{1}{2}\right) = \frac{m}{n}$, $m$ and $n$ are co-prime numbers, then $m + n$ is equal to _____.

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

If the solution curve \( y=y(x) \) of the differential equation \( y^{2}dx+(x^{2}-xy+y^{2})dy=0 \), which passes through the point \( (1,1) \) and intersects the line \( y=\sqrt{3}\,x \) at the point \( (\alpha,\sqrt{3}\alpha) \), then value of \( \log_{e}(\sqrt{3}\alpha) \) is equal to:

Let the slope of the tangent to a curve \( y=f(x) \) at \( (x,y) \) be given by \( 2 \tan x \left( \cos x - y \right) \). If the curve passes through the point \( \left( \frac{\pi}{4} , 0 \right) \), then the value of \( \int_{0}^{\pi/2} y \, dx \) is equal to:

Let \( y=y(x) \) be a differentiable function in the interval \( (0,\infty) \) such that \( y(1)=2 \), and \( \lim_{t\to x} \left(\dfrac{t^{2}y(x)-x^{2}y(t)}{x-t}\right)=3 \) for each \( x>0 \). Then \( 2y(2) \) 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 \( (x^{2} - 3y^{2})\,dx + 3xy\,dy = 0, \, y(1) = 1 \). Then \( 6y^{2}(e) \) is equal to:

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

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:

If the solution curve of the differential equation \( \dfrac{dy}{dx}=\dfrac{x+y-2}{x-y} \) passes through the points \( (2,1) \) and \( (k+1,2) \), \( k>0 \), then

Let \( y=y(x) \) be the solution of the differential equation \( \left(x+y+2\right)^{2} \, dx = dy, \; y(0)=-2 \). Let the maximum and minimum values of the function \( y=y(x) \) in \( \left[0,\tfrac{\pi}{3}\right] \) 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 ____________.

If the roots of the quadratic equation \( ax^2 + bx + 6 = 0 \) are equal, then minimum value of \( (6a - b + 2) \) is: (\( a, b \in \mathbb{R} \))

Let \( f:\mathbb{R}\to \mathbb{R} \) be a twice differentiable function such that the quadratic equation \( f(x)m^{2}-2f'(x)m+f''(x)=0 \) in \( m \) has two equal roots for every \( x\in\mathbb{R} \). If \( f(0)=1 \), \( f'(0)=2 \), and \( (\alpha,\beta] \) is the largest interval in which the function \( f\left(\log_{e}x - x\right) \) is increasing, then \( \alpha+\beta \) is equal to ________.

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

Let the solutions curve of the differential equation: \( x \dfrac{dy}{dx}-y=\sqrt{y^{2}+16x^{2}}, \ y(1)=3 \) be \( y=y(x) \). Then \( y(2) \) is equal to:

The differential equation of the family of circles passing through the points \( (0,2) \) and \( (0,-2) \) is:

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:

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

Let the solution curve \( y=y(x) \) of the differential equation \( (1+e^{2x})\!\left(\dfrac{dy}{dx}+y\right)=1 \) pass through the point \( \left(0,\dfrac{\pi}{2}\right) \). Then, \( \displaystyle\lim_{x\to \infty} e^{x}y(x) \) is equal to:

Let \( f \) be a differentiable function in the interval \( (0, \infty) \) such that \( f(1) = 1 \) and \( \lim_{t \to x} \frac{t^{2} f(x) - x^{2} f(t)}{t - x} = 1 \) for each \( x > 0 \). Then \( 2 f(2) + 3 f(3) \) is equal to ____________.

If \( f:\mathbb{R} \to \mathbb{R} \) be a differentiable function such that \( f'(x) + f(x) = \int_{0}^{2} f(t) \, dt \). If \( f(0) = e^{-2} \), then \( 2f(0) - f(2) \) 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 \( y=y(x) \) be the solution of the differential equation \( \dfrac{dy}{dx}=(1+x+x^{2})(1-y+y^{2}) \), \( y(0)=\dfrac{1}{2} \). Then \( (2y(1)-1) \) is equal to:

Let a conic \( C \) pass through the point \( (4,-2) \) and \( P(x,y) \), \( x \ge 3 \), by any point on \( C \). Let the slope of the line touching the conic \( C \) only at a single point \( P \) be half the slope of the line joining the points \( P \) and \( (3,-5) \). If the focal distance of the point \( (7,1) \) on \( C \) is \( d \), then \( 12d \) equals:

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:

Let the solution curve \( y = y(x) \) 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 \( (1,0) \) and \( (2\alpha,\alpha) \), \( \alpha > 0 \). Then \( \alpha \) is equal to:

If \( \sin\!\left( \dfrac{y}{x} \right) = \log_{e}|x| + \dfrac{\alpha}{2} \) is the solution of the differential equation \( x \cos\!\left( \dfrac{y}{x} \right) \dfrac{dy}{dx} = y \cos\!\left( \dfrac{y}{x} \right) + x \) and \( y(1) = \dfrac{\pi}{3} \), then \( \alpha^{2} \) is equal to:

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

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

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

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

The general solution of the differential equation \( (x-y^{2})dx+y(5x+y^{2})dy=0 \) is :

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

Let \( f \) be a twice differentiable function on \( \mathbb{R} \). If \( f'(0) = 4 \) and \( f(x) + \displaystyle\int\limits_{0}^{x} (x-t) f'(t) \, dt = (e^{2x} + e^{-2x}) \cos 2x + \dfrac{2}{a} x \), then \( (2a+1)^5 a^2 \) is equal to ____________ .

Let \( f \) be a differentiable function such that \(\displaystyle x^{2}f(x)-x = 4\!\int_{0}^{x} t f(t)\,dt \), \( f(1)=\dfrac{2}{3} \). Then \( 18f(3) \) is equal to:

A particle is moving in the \( xy \)-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 \( PQ \), then \( C \) is a parabola with:

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:

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

If \( y = y(x) \) is the solution of the differential equation, \(\displaystyle \sqrt{4 - x^2} \frac{dy}{dx} = \left( \left( \sin^{-1} \left( \frac{x}{2} \right) \right)^2 - y \right) \sin^{-1} \left( \frac{x}{2} \right) \), \( -2 \le x \le 2 \), \( \displaystyle y(2) = \frac{\pi^{2} - 8}{4} \), then \( y^{2}(0) \) is equal to _________.

Let \( f(x)=x-1 \) and \( g(x)=e^x \) for \( x \in \mathbb{R} \). If \( \frac{dy}{dx}=\Big(e^{-2\sqrt{x}} g(f(f(x))) - \frac{y}{\sqrt{x}}\Big) \), \( y(0)=0 \), then \( y(1) \) is:

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

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

Let the solution curve of the differential equation \( xdy = \left( \sqrt{x^2 + y^2} + y \right) dx, \ x > 0, \) intersect the line \( x = 1 \) at \( y = 0 \) and the line \( x = 2 \) at \( y = \alpha \). Then the value of \( \alpha \) is:

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

Let \( y = y(x) \) be a solution of the differential equation \( (x\cos x) \, dy + (xy\sin x + y\cos x - 1)\, dx = 0 \), \( 0 < x < \dfrac{\pi}{2} \). If \( \dfrac{\pi}{3} \, y\!\left(\dfrac{\pi}{3}\right) = \sqrt{3} \), then \( \left| \dfrac{\pi}{6}\, y''\!\left(\dfrac{\pi}{6}\right) + 2\, y'\!\left(\dfrac{\pi}{6}\right) \right| \) is equal to __________.

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

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

For the curve \( C:\ (x^{2}+y^{2}-3)+(x^{2}-y^{2}-1)^{5}=0 \), the value of \( 3y' - y^{3}y'' \), at the point \( (\alpha,\alpha) \), \( \alpha>0 \), on \( C \), is equal to ________.

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 \( x\!\left(e^{4}-2\right)=1 \). Then \( x\!\left(e^{9}-2\right) \) is equal to:

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:

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

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

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

If the solution curve of the differential equation \( (y-2\log_e x)\,dx+(x\log_e x^2)\,dy=0,\; x>1 \) passes through the points \( \left(e,\dfrac{4}{3}\right) \) and \( (e^4,\alpha) \), then \( \alpha \) is equal to ________.

Let \(\displaystyle \int_{0}^{x}\sqrt{1-\left(y'(t)\right)^{2}}\,dt=\int_{0}^{x}y(t)\,dt,\ 0\le x\le 3,\ y\ge 0,\ y(0)=0 \). Then at \( x=2 \), \( y''+y+1 \) is equal to

Let a smooth curve \( y=f(x) \) be such that the slope of the tangent at any point \( (x,y) \) on it is directly proportional to \( \left( \dfrac{-y}{x} \right) \). If the curve passes through the points \( (1,2) \) and \( (8,1) \) then \( \left| y\left(\dfrac{1}{8}\right) \right| \) is equal to:

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:

For a differentiable function \( f:\mathbb{R} \to \mathbb{R} \), suppose \( f'(x) = 3f(x) + \alpha \), where \( \alpha \in \mathbb{R} \), \( f(0) = 1 \) and \(\displaystyle \lim_{x \to -\infty} f(x) = 7 \). Then \( 9 f(\ln 3) \) is equal to ________.

Let \( f:\mathbb{R}\rightarrow \mathbb{R} \) be such that \( f(xy)=f(x)f(y) \), for all \( x,y\in \mathbb{R} \) and \( f(0)\neq 0 \). Let \( g:[1,\infty)\rightarrow \mathbb{R} \) be a differentiable function such that
\( x^{2}g(x)=\displaystyle\int_{1}^{x}\big(t^{2}f(t)-t\,g(t)\big)\,dt \)
Then \( g(2) \) is equal to:

Let the solution curve \( x=x(y) \), \( 0<y<\dfrac{\pi}{2}\), of the differential equation \((\log_e(\cos y))^2 \cos y \, dx - (1 + 3x \log_e(\cos y)) \sin y \, dy = 0\) satisfy \(x\left(\dfrac{\pi}{3}\right) = \dfrac{1}{2\log_e 2}\).If \(\displaystyle x\left(\dfrac{\pi}{6}\right) = \dfrac{1}{\log_e m - \log_e n}\),where \(m\) and \(n\) are coprime, then \(mn\) is equal to:

If the solution curve \( y=y(x) \) of the differential equation \( (1+y^{2})(1+\log_{e} x)\,dx + x\,dy = 0,\ x>0 \) passes through the point \( (1,1) \) and \( y(e)=\frac{\alpha-\tan\!\left(\frac{3}{2}\right)}{\beta+\tan\!\left(\frac{3}{2}\right)} \), then \( \alpha+2\beta \) is ____________.

If \( y = y(x) \) is the solution of the differential equation \( \dfrac{dy}{dx} + 2y = \sin(2x), \ y(0) = \dfrac{3}{4} \), then \( y\!\left(\dfrac{\pi}{8}\right) \) is equal to:

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

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

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:

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

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 \( f:\mathbb{R}\rightarrow \mathbb{R} \) be a thrice differentiable odd function satisfying \( f'(x)\ge 0 \), \( f''(x)=f(x) \), \( f(0)=0 \), \( f'(0)=3 \). Then \( 9\,f(\log_{e}3) \) is equal to __________.

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

Let \( f \) be a differentiable function such that \( 2(x+2)^{2} f(x) - 3(x+2)^{2} = 10 \int_{0}^{x} (t+2) f(t) \, dt, \ x \ge 0 \). Then \( f(2) \) is equal to ________.

If \( y=y(x) \) is the solution of the differential equation \( (1+e^{2x})\frac{dy}{dx}+2(1+y^{2})e^{x}=0 \) and \( y(0)=0 \), then \( 6\left(y'(0)+y(\log_{e}\sqrt{3})\right)^{2} \) is equal to:

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:

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:

If the solution curve \( f(x,y)=0 \) of the differential equation \( (1+\log_{e}x)\dfrac{dx}{dy}-x\log_{e}x=e^{y},\ x>0 \) passes through the points \( (1,0) \) and \( (\alpha,2) \), then \( \alpha^{\alpha} \) is equal to:

Let \( y=y(x) \) be the solution curve of the differential equation \( \dfrac{dy}{dx}+\dfrac{1}{x^2-1}y=\left(\dfrac{x-1}{x+1}\right)^{1/2},\ x>1 \) passing through the point \( \left(2,\sqrt{\dfrac{1}{3}}\right) \). Then \( \sqrt{7}\,y(8) \) 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 \( y = y(x) \) be the solution of the differential equation \( (x^{2} + 4)^{2} \, dy + (2x^{3}y + 8xy - 2) \, dx = 0 \).
If \( y(0) = 0 \), then \( y(2) \) 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 \( y=y(t) \) be a solution of the differential equation \( \frac{dy}{dt}+\alpha y=\gamma e^{-\beta t} \). Where, \( \alpha>0 \), \( \beta>0 \) and \( \gamma>0 \). Then \( \displaystyle \lim_{t\to\infty} y(t) \).

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

Let the solution curve \( y = y(x) \) of the differential equation \( (4 + x^2)\,\,dy - 2x(x^2 + 3y + 4)\,dx = 0 \) pass through the origin. Then \( y(2) \) is equal to ________________ .

If \( y=y(x) \) is the solution curve of the differential equation \( \dfrac{dy}{dx}+y\tan x = x\sec x \), \( 0 \le x \le \dfrac{\pi}{3} \), \( y(0)=1 \)
Then \( y\!\left(\dfrac{\pi}{6}\right) \) is equal to:

Let the tangent at any point \( P \) on a curve passing through the points \( (1, 1) \) and \( \left(\dfrac{1}{10}, 100\right) \), intersect positive \( x \)-axis and \( y \)-axis at the points \( A \) and \( B \) respectively. If \( PA : PB = 1 : k \) and \( y(x) \) is the solution of the differential equation \( e^{\dfrac{dy}{dx}} = kx + \dfrac{k}{2},\ y(0) = k, \) then \( 4y(1) - 4\log_{e} 3 \) is equal to ______.

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

Let \( y = y(x) \) be the solution curve of the differential equation \( \dfrac{dy}{dx} = \dfrac{y}{x}\big(1 + xy^{2}(1 + \log_{e} x)\big) \), \( x > 0 \), \( y(1) = 3 \). Then \( \dfrac{y^{2}(x)}{9} \) 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 \( y = y(x) \) be the solution of the differential equation \( x \log_e x \, \frac{dy}{dx} + y = x^2 \log_e x, \, (x > 1) \). If \( y(2) = 2 \), then \( y(e) \) is equal to:

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

Let \( y=y(x) \) be the solution of the differential equation \( \dfrac{dy}{dx}+3(\tan^{2}x)\,y+3y=\sec^{2}x, \; y(0)=\dfrac{1}{3}+e^{3} \). Then \( y\!\left(\dfrac{\pi}{4}\right) \) 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?

Let the solution \( y = y(x) \) of the differential equation \( \frac{dy}{dx} - y = 1 + 4\sin x \) satisfy \( y(\pi) = 1 \). Then \( y\!\left(\frac{\pi}{2}\right) + 10 \) is equal to __________.

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 \( (x^2+1)\,y' - 2xy = (x^4 + 2x^2 + 1)\cos x, \; y(0)=1 \).Then \(\displaystyle \int_{-3}^{3} y(x)\,dx \) is:

Let \( y = f(x) \) be the solution of the differential equation \(\displaystyle \frac{dy}{dx} + \frac{xy}{x^2 - 1} = \frac{x^6 + 4x}{\sqrt{1 - x^2}},\ -1 < x < 1 \) such that \( f(0) = 0 \). If \(\displaystyle 6 \int_{-1/2}^{1/2} f(x)\, dx = 2\pi - \alpha \) then \( \alpha^2 \) is equal to __________.

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

Suppose the solution of the differential equation \( \frac{dy}{dx}=\frac{(2+\alpha)x-\beta y+2}{\beta x-2\alpha y-(\beta y-4\alpha)} \) represents a circle passing through origin. Then the radius of this circle is:

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

Let \( \frac{dy}{dx} = \frac{ax - by + a}{bx + cy + a} \), where \( a, b, c \) are constants, represent a circle passing through the point \( (2,5) \). Then the shortest distance of the point \( (11,6) \) from this circle is:

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:

If \(\displaystyle \frac{dx}{dy} = \frac{1 + x - y^2}{y} , \; x(1) = 1 \), then \( 5x(2) \) 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:

If \( x=x(t) \) is the solution of the differential equation \( (t+1)\,dx=(2x+(t+1)^{4})\,dt \), \( x(0)=2 \), then, \( x(1) \) equals ______.

Let \( f:[1,\infty)\to \)(\(\mathbb{R}\)) be a differentiable function. If \( 6\int_{1}^{x} f(t)\,dt=3x f(x)+x^{3}-4 \) for all \( x\ge 1 \), then the value of \( f(2)-f(3) \) is:

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JEE Main 2023 29th Jan Morning Shift

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