If $\lim _{x \rightarrow \infty} y(x)=\frac{\pi}{2}$,then the solution of $x^3 \sin y \frac{d y}{d x}=2$ is $\cos y=$

If the solution curve $y=y(x)$ of the differential equation $(1+y^2)(1+\log_e x) dx + x dy = 0, x>0$ passes through the point $(1,1)$ and $y(e) = \frac{\alpha-\tan(3/2)}{\beta+\tan(3/2)}$,then $\alpha+2\beta$ is equal to:

If the curve $y = f(x)$ passes through the point $(1, e)$ and satisfies the differential equation $dy = y(2 + \log_e x) dx, x > 0$,then $f(e)$ is equal to:

Let $y=y(x)$ be the solution of the differential equation $\log _{e}\left(\frac{d y}{d x}\right)=3 x+4 y$,with $y(0)=0$. If $y\left(-\frac{2}{3} \log _{e} 2\right)=\alpha \log _{e} 2$,then the value of $\alpha$ is equal to:

The solution of $e^{dy/dx} = x+1, y(0) = 3$ is

Find a particular solution satisfying the given condition: $\cos \left(\frac{dy}{dx}\right) = a$ $(a \in R)$; $y = 1$ when $x = 0$.