The general solution of the differential equation $\frac{dy}{dx} + \frac{y \ln y}{x} = \frac{y(\ln y)^2}{x^2}$ is (where $C$ is an arbitrary constant):

The differential equation $\frac{dy}{dx} = \frac{1}{ax + by + c}$,where $a, b, c$ are all non-zero real numbers,is

The general solution of the differential equation $\frac{dy}{dx} + y g'(x) = g(x) g'(x)$ is

Let $f:[1, \infty) \rightarrow [2, \infty)$ be a differentiable function such that $f(1)=2$. If $6 \int_1^x f(t) dt = 3x f(x) - x^3$ for all $x \geq 1$,then the value of $f(2)$ is

Find a particular solution satisfying the given condition: $\frac{dy}{dx} + 2y \tan x = \sin x$; $y = 0$ when $x = \frac{\pi}{3}$.

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 $.....$.