The solution of the equation $(x-4y^3) \frac{dy}{dx}-y=0, (y>0)$ is

Let $y=y(x)$ be the solution of the differential equation $e^{y} \frac{d y}{d x}-2 e^{y} \sin x+\sin x \cos ^{2} x=0$,with $y(\frac{\pi}{2})=0$. If $y(0)=\log _{e}(\alpha+\beta e^{-2})$,then $4(\alpha+\beta)$ is equal to $....$

Let $f : R \rightarrow R$ be a differentiable function such that $f^{\prime}(x)+f(x)=\int \limits_0^2 f(t) dt$. If $f(0)=e^{-2}$,then $2f(0)-f(2)$ is equal to $.........$.

If the solution of the differential equation $x y^{\prime}=y+x^2 \sin x$ subject to the condition $y(\pi)=0$ is $y=f(x)$ and $f(x)$ has an extreme value at $x=\alpha$, then

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

The general solution of $\cos ^2 x \frac{d y}{d x}+y=\tan x$ is