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

The general solution of $\left(x \frac{dy}{dx} - y\right) \sin \frac{y}{x} = x^3 e^x$ is

If the solution $y=y(x)$ of the differential equation $(x^4+2x^3+3x^2+2x+2)dy-(2x^2+2x+3)dx=0$ satisfies $y(-1)=-\frac{\pi}{4}$,then $y(0)$ is equal to:

The general solution of the differential equation $\frac{dy}{dx} = \cos^2(3x+y)$ is $\tan^{-1}\left(\frac{\sqrt{3}}{2} \tan(3x+y)\right) = f(x)$. Then,$f(x) =$

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

Let $y=y(x)$ be a solution of the differential equation,$\sqrt{1-x^{2}} \frac{dy}{dx}+\sqrt{1-y^{2}}=0, |x| < 1$. If $y\left(\frac{1}{2}\right)=\frac{\sqrt{3}}{2}$,then $y\left(\frac{-1}{\sqrt{2}}\right)$ is equal to