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

The general solution of the differential equation $\frac{dy}{dx} = \frac{x+2y-1}{x+2y+1}$ is

The solution of the differential equation $\sqrt{a + x} \frac{dy}{dx} + xy = 0$ is

The particular solution of the differential equation $\log\left(\frac{dy}{dx}\right) = x$,when $x = 0, y = 1$ is .....

$A$ particular solution of $3 e^x \tan y \, dx + (1 - e^x) \sec^2 y \, dy = 0$ with $y(1) = \frac{\pi}{4}$ 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: