If $\int x f(x) dx + \frac{f(x)}{2} = 0$, then $f(x)$ is equal to

If $y=y(x)$ satisfies the differential equation $16(\sqrt{x+9\sqrt{x}})(4+\sqrt{9+\sqrt{x}}) \cos y \, dy = (1+2 \sin y) \, dx$ for $x > 0$, and $y(256)=\frac{\pi}{2}$, $y(49)=\alpha$, then $2 \sin \alpha$ is equal to:

The slope of a curve at any point is inversely proportional to twice the $y$-coordinate of that point. If the curve passes through $(4, 3)$,then the equation of the curve is:

The particular solution of the differential equation $(1+e^{2x}) dy + e^x(1+y^2) dx = 0$ at $x=0$ and $y=1$ is

The solution of the differential equation $e^{2y} (1 + \ln x)dx + \csc y (2 + \cot y)dy = 0$ satisfying $y(1) = \frac{\pi}{2}$ is

The solution of the differential equation $\frac{dx}{x} + \frac{dy}{y} = 0$ is