Variable separable

Let $y:(-\infty,\infty)\to(0,\infty)$ be the solution of the differential equation
$$\frac{dy}{dx}=\frac{e^{5x}y^3+y^3}{e^x+e^xy^4},$$
satisfying $y(0)=\frac1{\sqrt2}$. Then the value of $y(\log_e2)$ is

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

Let $y=y(x)$ be the solution of the differential equation $\dfrac{dy}{dx}=\left(1+x+x^2\right)\left(1-y+y^2\right)$, $y(0)=\dfrac12$. Then $(2y(1)-1)$ is equal to

Let $f$ be a twice differentiable function such that $f(x)=\displaystyle\int_0^x\tan(t-x)\,dt-\int_0^xf(t)\tan t\,dt$, $x\in\left(-\dfrac\pi2,\dfrac\pi2\right)$. Then $f''\left(\dfrac\pi6\right)+12f'\left(-\dfrac\pi6\right)+f\left(\dfrac\pi6\right)$ is equal to ________.

For all $x > 0$, let $y_1(x)$, $y_2(x)$, and $y_3(x)$ be the functions satisfying

$$\begin{aligned} &\frac{dy_1}{dx} - (\sin x)^2\,y_1 = 0, & y_1(1) &= 5,\\ &\frac{dy_2}{dx} - (\cos x)^2\,y_2 = 0, & y_2(1) &= \frac13,\\ &\frac{dy_3}{dx} - \left(\frac{2 - x^3}{x^3}\right)y_3 = 0, & y_3(1) &= \frac{3}{5e}, \end{aligned}$$

respectively. Then

$$\lim_{x\to0^+}\frac{y_1(x)\,y_2(x)\,y_3(x) + 2x}{e^{3x}\sin x}$$

is equal to _____.

If $y(x)$ is the solution of the differential equation
$$x\,dy - (y^2 - 4y)\,dx = 0 \quad\text{for } x > 0, \qquad y(1) = 2,$$
and the slope of the curve $y = y(x)$ is never zero, then the value of $10\,y(\sqrt2)$ is _____.

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

Let $S=(0,2\pi)-\left\{\dfrac\pi2,\dfrac{3\pi}4,\dfrac{3\pi}2,\dfrac{7\pi}4\right\}$. Let $y=y(x)$, $x\in S$, be the solution curve of the differential equation $\dfrac{dy}{dx}=\dfrac1{1+\sin2x}$, $y\left(\dfrac\pi4\right)=\dfrac12$. If the sum of abscissas of all the points of intersection of the curve $y=y(x)$ with the curve $y=\sqrt2\sin x$ is $\dfrac{k\pi}{12}$, then $k$ is equal to ______.

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 $y=y(x)$ is the solution of the differential equation $\left(1+e^{2x}\right)\dfrac{dy}{dx}+2\left(1+y^2\right)e^x=0$ and $y(0)=0$, then $6\left(y'(0)+\left(y\left(\log_e\sqrt3\right)\right)^2\right)$ is equal to

A hostel has 100 students. On a certain day (consider it day zero) it was found that two students are infected with some virus. Assume that the rate at which the virus spreads is directly proportional to the product of the number of infected students and the number of non-infected students. If the number of infected students on 4$^{\text{th}}$ day is 30, then number of infected students on 8$^{\text{th}}$ day will be ______.

The slope of the tangent to a curve $C:y=y(x)$ at any point $(x,y)$ on it is $\dfrac{2e^{2x}-6e^{-x}+9}{2+9e^{-2x}}$. If C passes through the points $\left(0,\dfrac12+\dfrac\pi{2\sqrt2}\right)$ and $\left(\alpha,\dfrac12e^{2\alpha}\right)$, then $e^\alpha$ 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 $\left|y\left(\dfrac18\right)\right|$ is equal to

Let $f$ be a differentiable function satisfying $f(x)=\dfrac2{\sqrt3}\displaystyle\int_0^{\sqrt3}f\left(\frac{\lambda^2x}3\right)d\lambda$, $x>0$ and $f(1)=\sqrt3$. If $y=f(x)$ passes through the point $(\alpha,6)$, then $\alpha$ is equal to ______.

Let the area enclosed by the solution curve $y=y(x)$ of the differential equation $\dfrac{dy}{dx}+\dfrac{x-2}{y-b}=2$, b > 0, y(5) = 0 is 13π. Let $y=y(x)$ intersect the y-axis at the points P and Q. If the tangents to $y=y(x)$ at P and Q meet at T, then the area of ΔPTQ is

The population P = P(t) at time 't' of a certain species follows the differential equation $\dfrac{dP}{dt}=0.5P-450$. If P(0) = 850, then the time at which population becomes zero is :

Let $b$ be a nonzero real number. Suppose $f:\mathbb R\to\mathbb R$ is a differentiable function such that $f(0) = 1$. If the derivative $f'$ of $f$ satisfies the equation

$$f'(x) = \frac{f(x)}{b^2 + x^2}$$

for all $x\in\mathbb R$, then which of the following statements is/are TRUE?

Let $\Gamma$ denote a curve $y = y(x)$ which is in the first quadrant and let the point $(1, 0)$ lie on it. Let the tangent to $\Gamma$ at a point $P$ intersect the $y$-axis at $Y_P$. If $PY_P$ has length 1 for each point $P$ on $\Gamma$, then which of the following options is/are correct?

Let $f: \mathbb{R} \to \mathbb{R}$ and $g: \mathbb{R} \to \mathbb{R}$ be two non-constant differentiable functions. If

$$f'(x) = \left(e^{(f(x) - g(x))}\right) g'(x) \quad \text{for all } x \in \mathbb{R},$$

and $f(1) = g(2) = 1$, then which of the following statement(s) is (are) TRUE?

Let $f: \mathbb{R} \to \mathbb{R}$ be a differentiable function with $f(0) = 0$. If $y = f(x)$ satisfies the differential equation

$$\frac{dy}{dx} = (2 + 5y)(5y - 2),$$

then the value of $\displaystyle\lim_{x \to -\infty} f(x)$ is _____.