Let $y = y(x)$ be the solution of the differential equation,$x\frac{dy}{dx} + y = x \ln x$,for $x > 1$. If $2y(2) = \ln 4 - 1$,then $y(e)$ is equal to

Let the solution curve $y=y(x)$ of the differential equation $(1+e^{2x})(\frac{dy}{dx}+y)=1$ pass through the point $(0, \frac{\pi}{2})$. Then,$\lim_{x \rightarrow \infty} e^{x} y(x)$ is equal to.

$A$ function $y = f(x)$ satisfies $(x + 1)f'(x) - 2(x^2 + x)f(x) = \frac{e^{x^2}}{(x + 1)}$. If $f(0) = 5$,then $f(x)$ is:

An integrating factor of the equation $(1+y+x^2 y) dx+(x+x^3) dy=0$ is

Let $g$ be a differentiable function such that $\int_0^x g(t) dt = x - \int_0^x tg(t) dt$ for $x \geq 0$. Let $y = y(x)$ satisfy the differential equation $\frac{dy}{dx} - y \tan x = 2(x+1) \sec x g(x)$ for $x \in [0, \frac{\pi}{2})$. If $y(0) = 0$,then $y(\frac{\pi}{3})$ is equal to

Let $y=y(x)$ be the solution of the differential equation $(\tan x)^{1/2} dy = (\sec^3 x - (\tan x)^{3/2} y) dx$, where $0 < x < \frac{\pi}{2}$ and $y(\frac{\pi}{4}) = \frac{6\sqrt{2}}{5}$. If $y(\frac{\pi}{3}) = \frac{4}{5}\alpha$, then $\alpha^4$ equals . . . . . . .