$A$ curve is such that the area of the region bounded by the coordinate axes,the curve,and the ordinate of any point on it is equal to the cube of that ordinate. The curve represents:

The differential equation $\frac{dy}{dx} = \frac{\sqrt{1-y^2}}{y}$ determines a family of circles with

The differential equation $y\frac{dy}{dx} + x = a$ ($a$ is any constant) represents:

The slope of the tangent to a curve $C : y = y(x)$ at any point $(x, y)$ on it is $\frac{2e^{2x} - 6e^{-x} + 9}{2 + 9e^{-2x}}$. If $C$ passes through the points $(0, \frac{1}{2} + \frac{\pi}{2\sqrt{2}})$ and $(\alpha, \frac{1}{2}e^{2\alpha})$,then $e^{\alpha}$ is equal to:

If the solution of the differential equation $\frac{dy}{dx} = \frac{1+x}{2y}$ is a conic passing through the point $(1, 1)$,then its eccentricity is:

An ice ball melts at a rate proportional to the amount of ice present at that instant. Half of the initial quantity of ice melts in $15 \text{ minutes}$. Let $x_0$ be the initial quantity of ice. If after $30 \text{ minutes}$ the amount of ice left is $k x_0$,then the value of $k$ is: