The rate of change of volume of a spherical balloon at any instant is directly proportional to its surface area. If initially its radius is $3 \ cm$,and after $2 \ minutes$ its radius becomes $9 \ cm$,then what is the radius of the balloon after $4 \ minutes$ (in $cm$)?

Given $\frac{d^2 y}{d x^2}+\cot x \frac{d y}{d x}+4 y \operatorname{cosec}^2 x=0$. Changing the independent variable $x$ to $z$ by the substitution $z=\log \tan \frac{x}{2}$, the equation is changed to

The differential equation of displacement of all $Simple \ Harmonic \ Motions$ of given period $T = 2\pi / n$ is:

The solution of the differential equation $x \frac{d^2 y}{d x^2} = 1$ at $x = y = 1$ with $\frac{dy}{d x} = 0$ at $x = 1$ is:

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

If a body cools from $135^{\circ} C$ to $80^{\circ} C$ at a room temperature of $25^{\circ} C$ in $60 \text{ minutes}$,then the temperature of the body after $2 \text{ hours}$ is: (in $^{\circ} C$)