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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:

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

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

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 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 \( y = y(x) \) 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:

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:

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:

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:

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

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 ____________.

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 ____________.

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:

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 \( \dfrac{dy}{dx}=2x(x+y)^{3}-x(x+y)-1 \), \( y(0)=1 \). Then, \( \left(\dfrac{1}{\sqrt{2}}+y\!\left(\dfrac{1}{\sqrt{2}}\right)\right)^{2} \) equal:

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

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 ________.

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 \( y = y(x) \) 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:

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 \( f(x) \) 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 \( y = y(x) \) 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 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 __________.

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:

If \(\displaystyle \frac{dx}{dy} = \frac{1 + x - y^2}{y} , \; x(1) = 1 \), then \( 5x(2) \) is equal to \( \_\_\_\_\_\_\_\_\_\_ \).

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 ______.

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JEE Main 2024 8th April Morning Shift

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: