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

If $\frac{dy}{dx} = y + 3 > 0$ and $y(0) = 2$,then $y(\ln 2)$ is equal to:

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 general solution of the differential equation $\frac{dy}{dx} = \frac{x+y-3}{x+y-7}$ is

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 the differential equation $e^{\frac{dy}{dx}} = x+1$ with the initial condition $y(0) = 5$ for $x \in (-1, \infty)$ is: