The general solution of the differential equation $\cos x \sin y \, dx + \sin x \cos y \, dy = 0$ is

The solution of the differential equation $2x \frac{dy}{dx} - y = 0$,with the initial condition $y(1) = 2$,represents which of the following curves?

The general solution of the differential equation $\frac{dy}{dx} = \cot x \cdot \cot y$ is

Let $y = y(x)$ be the solution curve of the differential equation $(1 + \sin x) \frac{dy}{dx} + (y + 1) \cos x = 0$ with the condition $y(0) = 0$. If the curve $y = y(x)$ passes through the point $(\alpha, -\frac{1}{2})$, then a value of $\alpha$ is:

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

If a curve passes through the point $\left( 2, \frac{7}{2} \right)$ and has slope $\left( 1 - \frac{1}{x^2} \right)$ at any point $(x, y)$ on it,then the ordinate of the point on the curve whose abscissa is $-2$ is