If $\frac{dy}{dx} = e^{-2y}$ and $y = 0$ when $x = 5$,the value of $x$ for $y = 3$ 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 $\frac{dy}{dx} = \cos(x + y)$ is:

The general solution of the differential equation $e^{\frac{1}{2}\left(\frac{dy}{dx}\right)}=3^x$ is (where $C$ is a constant of integration).

Let a smooth curve $y=f(x)$ be such that the slope of the tangent at any point $(x, y)$ on it is directly proportional to $\left(\frac{-y}{x}\right)$. If the curve passes through the points $(1, 2)$ and $(8, 1)$,then $\left| y \left(\frac{1}{8}\right) \right|$ is equal to

If a curve passes through the point $\left( 2, \frac{7}{2} \right)$ and has slope $\left( 1 - \frac{1}{x^2} \right)$ at any point $(x, y)$ on it,then the ordinate of the point on the curve whose abscissa is $-2$ is