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

The solution of $(\text{cosec } x \log y) dy + (x^2 y) dx = 0$ is

The equation of the curve passing through $(3, 9)$ which satisfies the differential equation $\frac{dy}{dx} = x + \frac{1}{x^2}$ is

The solution of ${e^{2x - 3y}}dx + {e^{2y - 3x}}dy = 0$ is

The general solution of the differential equation $\tan x \tan y \, dx + \cos^2 x \operatorname{cosec}^2 y \, dy = 0$ 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 $.....$