Number of straight lines which satisfy the differential equation $\frac{dy}{dx} + x \left( \frac{dy}{dx} \right)^2 - y = 0$ is:

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

$A$ spherical rain drop evaporates at a rate proportional to its surface area. If initially its radius is $3 \ mm$ and after $1 \ second$ it is reduced to $2 \ mm$,then at any time $t$ its radius is (where $0 \leq t < 3$)

The rate of growth of bacteria in a culture is proportional to the number of bacteria present and the bacteria count is $1000$ at initial time $t = 0$. The number of bacteria is increased by $20\%$ in $2$ hours. If the population of bacteria is $2000$ after $\frac{k}{\log_{e}\left(\frac{6}{5}\right)}$ hours,then $\left(\frac{k}{\log_{e} 2}\right)^{2}$ is equal to

$A$ normal is drawn at a point $P(x, y)$ of a curve $y=f(x)$. The normal meets the $X$-axis at $Q$. If the length $l(PQ) = k$,where $k$ is a constant,find the equation of the curve passing through $(0, k)$.