If $y=y(x)$ is the solution of the differential equation $\left(\frac{5+e^x}{2+y}\right) \frac{dy}{dx}+e^x=0$ satisfying $y(0)=1$,then a value of $y(\log 13)$ is

The solution of $(x+y)^{2} \frac{dy}{dx} = a^{2}$ (where $a$ is a constant) is:

Let a curve $y=y(x)$ be given by the solution of the differential equation $\cos \left(\frac{1}{2} \cos ^{-1}\left(e^{-x}\right)\right) d x=\sqrt{e^{2 x}-1} \,d y$. If it intersects the $y$-axis at $y=-1$,and the intersection point of the curve with the $x$-axis is $(\alpha, 0)$,then $e^{\alpha}$ is equal to $.....$

The solution of the differential equation $\frac{dy}{dx} = \frac{x-2y+1}{2x-4y}$ is

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

The general solution of the differential equation $\frac{d y}{d x}+\frac{\sin (2 x+y)}{\cos x}+2=0$ is: