The integrating factor of the differential equation $3 x \log _{e} x \frac{d y}{d x}+y=2 \log _{e} x$ is given by

Let $f$ be a differentiable function with $\lim_{x \rightarrow \infty} f(x) = 0$. If $y^{\prime} + y f^{\prime}(x) - f(x) f^{\prime}(x) = 0$ and $\lim_{x \rightarrow \infty} y(x) = 0$,then (where $y^{\prime} = \frac{dy}{dx}$):

Let $y=y(x)$ be the solution of the differential equation $\frac{dy}{dx}+2y \sec^2 x = 2 \sec^2 x + 3 \tan x \cdot \sec^2 x$ such that $y(0)=\frac{5}{4}$. Then $12\left(y\left(\frac{\pi}{4}\right)-e^{-2}\right)$ is equal to . . . . . . .

The solution of $\frac{d y}{d x}+\frac{1}{x}=\frac{e^y}{x^2}$ is

The solution of the differential equation $(x + 2y^3) \frac{dy}{dx} = y$ is

The solution of $\cos y + (x \sin y - 1) \frac{dy}{dx} = 0$ is