Let the solution curve $y=y(x)$ of the differential equation $(4+x^{2}) dy - 2x(x^{2}+3y+4) dx = 0$ pass through the origin. Then $y(2)$ is equal to

If $y=f(x)$ is the solution of the differential equation $(1+\cos^2 x) f'(x) - f(x) \sin 2x = 4 \sin 2x$ with $f(0)=0$, then $f(\frac{\pi}{3})=$

The integrating factor of $\frac{dy}{dx} + \frac{y}{x} = x^3 - 3$ is

If $f(x)$ is a differentiable function in the interval $(0, \infty)$ such that $f(1) = 1$ and $\mathop {\lim }\limits_{t \to x} \frac{{{t^2}f(x) - {x^2}f(t)}}{{t - x}} = 1$ for each $x > 0$,then $f(\frac{3}{2})$ is equal to

The integrating factor of the differential equation $(2x + 3y^2) dy = y dx$ $(y > 0)$ is

If $y = \left(\frac{2}{\pi} x - 1\right) \operatorname{cosec} x$ is the solution of the differential equation $\frac{dy}{dx} + p(x) y = \frac{2}{\pi} \operatorname{cosec} x$ for $0 < x < \frac{\pi}{2}$,then the function $p(x)$ is equal to