The integrating factor of the differential equation $\frac{dy}{dx} + (3x^2 \tan^{-1} y - x^3)(1 + y^2) = 0$ is

If the solution curve $y=f(x)$ of the differential equation $(x^{2}-4)y^{\prime}-2xy+2x(4-x^{2})^{2}=0$ for $x>2$ passes through the point $(3, 15)$, then the local maximum value of $f$ is:

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 $.....$.

Find the general solution of the differential equation: $x \frac{dy}{dx} + y - x + xy \cot x = 0$ $(x \neq 0)$.

The general solution of the differential equation $\frac{dy}{dx} + \frac{y}{x} = x^2 + 5$ is ....

An integrating factor of the differential equation $x \frac{dy}{dx} + y \log x = x e^x x^{-\frac{1}{2} \log x}$,$(x > 0)$ is