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

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

Find a particular solution satisfying the given condition: $\frac{dy}{dx} = y \tan x$; $y = 1$ when $x = 0$.

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:

If $\frac{dy}{dx} = \frac{2^x y + 2^y \cdot 2^x}{2^x + 2^{x+y} \log_e 2}$ and $y(0) = 0$,then for $y = 1$,the value of $x$ lies in the interval:

The general solution of the differential equation $\log \left(\frac{dy}{dx}\right) = ax + by$ is