Differential Equations

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

Let $y=f(x)$ be the real valued function defined on the interval $(0,\infty)$, satisfying $y(1)=0$ and the differential equation
$$x\frac{dy}{dx}=y-x^3.$$
Then which of the following statements is (are) TRUE ?

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

Let $x=x(y)$ be the solution of the differential equation $2y^2\dfrac{dx}{dy}-2xy+x^2=0$, $y>1$, $x(e)=e$. Then $x\left(e^2\right)$ 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 $y=y(x)$ be the solution of the differential equation:
$$\frac{dy}{dx}+\left(\frac{6x^2+\left(3x^2+2x^3+4\right)e^{-2x}}{\left(x^3+2\right)\left(2+e^{-2x}\right)}\right)y=2+e^{-2x},$$
$x\in(-1,2)$, satisfying $y(0)=\dfrac32$. If $y(1)=\alpha\left(2+e^{-2}\right)$, then $\alpha$ 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 ________.

Let $y=y(x)$ be the solution of the differential equation $x\sin\left(\dfrac yx\right)dy=\left(y\sin\left(\dfrac yx\right)-x\right)dx$, $y(1)=\dfrac\pi2$ and let $\alpha=\cos\left(\dfrac{y\left(e^{12}\right)}{e^{12}}\right)$. 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 $r\le6$, is ________.

Let $f:[1,\infty)\to\mathbf R$ be a differentiable function defined as $f(x)=\displaystyle\int_1^xf(t)\,dt+(1-x)(\log_ex-1)+e$. Then the value of $f(f(1))$ is :

Let $y=y(x)$ be the solution of the differential equation
$$(\tan x)^{1/2}\,dy=\left(\sec^3x-(\tan x)^{3/2}\,y\right)dx,\quad 0<x<\frac\pi2,\quad y\left(\frac\pi4\right)=\frac{6\sqrt2}5.$$
If $y\left(\dfrac\pi3\right)=\dfrac45\alpha$, then $\alpha^4$ equals ________.

Let $y=y(x)$ be the solution of the differential equation $\left(x^2-x\sqrt{x^2-1}\right)dy+\left(y\left(x-\sqrt{x^2-1}\right)-x\right)dx=0$, $x\ge1$. If $y(1)=1$, then the greatest integer less than $y\left(\sqrt5\right)$ is ________.

Let $f:\mathbf R\to\mathbf R$ be such that $f(xy)=f(x)f(y)$, for all $x,y\in\mathbf R$ and $f(0)\ne0$. Let $g:[1,\infty)\to\mathbf R$ be a differentiable function such that
$$x^2g(x)=\int_1^x\left(t^2f(t)-tg(t)\right)dt.$$
Then $g(2)$ is equal to :

Let $y=y(x)$ be the solution of the differential equation $x\sqrt{1-x^2}\,dy+\left(y\sqrt{1-x^2}-x\cos^{-1}x\right)dx=0$, $x\in(0,1)$, $\displaystyle\lim_{x\to1^-}y(x)=1$. Then $y\left(\dfrac12\right)$ equals :

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 _____.

Let $y(x)$ be the solution of the differential equation

$$x^2\frac{dy}{dx} + xy = x^2 + y^2,\qquad x > \frac1e,$$

satisfying $y(1) = 0$. Then the value of $2\dfrac{(y(e))^2}{y(e^2)}$ is _____.

Let $f(x)$ be a continuously differentiable function on the interval $(0,\infty)$ such that $f(1) = 2$ and

$$\lim_{t\to x}\frac{t^{10}f(x) - x^{10}f(t)}{t^9 - x^9} = 1$$

for each $x > 0$. Then, for all $x > 0$, $f(x)$ is equal to

Let $f : [1,\infty) \to \mathbb{R}$ be a differentiable function such that $f(1) = \dfrac13$ and $3\displaystyle\int_1^x f(t)\,dt = x\,f(x) - \dfrac{x^3}{3}$, $x \in [1,\infty)$. Let $e$ denote the base of the natural logarithm. Then the value of $f(e)$ is

For $x \in \mathbb{R}$, let $y(x)$ be a solution of the differential equation $(x^2-5)\dfrac{dy}{dx} - 2xy = -2x(x^2-5)^2$ such that $y(2) = 7$. Then the maximum value of the function $y(x)$ is _____.

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 _____.