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 :

Find the solution of the differential equation $\frac{dy}{dx} = \frac{1}{xy(x^2 \sin y^2 + 1)}$,where $C$ is the integral constant.

Suppose $f(x)$ is a differentiable real function such that $f(x) + f'(x) \le 1$ for all $x$ and $f(0)=0$. The largest possible value of $f(1)$ is

Let $f :(0, \infty) \rightarrow R$ be a function which is differentiable at all points of its domain and satisfies the condition $x^2 f^{\prime}(x)=2 x f(x)+3$,with $f(1)=4$. Then $2 f(2)$ is equal to:

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

The integrating factor of the differential equation $\frac{dy}{dx} + \frac{y}{x} = x^2$ is . . . . . . .