If $y = e^{-mx}$ is a solution of the differential equation $\frac{d^2y}{dx^2} + 4\frac{dy}{dx} + 3y = 0$, then the values of $m$ are

The rate of increase of the population of a country is proportional to the number present. If the population doubles in $50$ years,then the time taken by it to become four times itself is (in $years$)

The number of differentiable functions $y: (-\infty, \infty) \rightarrow [0, \infty)$ satisfying $y^{\prime} = 2 \sqrt{y}$ and $y(0) = 0$ is

The decay rate of radium is proportional to the amount present at any time $t$. If initially $60 \text{ gms}$ was present and half-life period of radium is $1600 \text{ years}$, then the amount of radium present after $3200 \text{ years}$ is (in $\text{ grams}$)

If the general solution of the differential equation $y' = \frac{y}{x} + \phi \left( \frac{x}{y} \right)$,for some function $\phi$,is given by $y \ln |cx| = x$,where $c$ is an arbitrary constant,then $\phi(2)$ is equal to:

Any curve passes through the point $(3, -4)$. If the slope of the tangent to the curve at any point $(x, y)$ is $\frac{2y}{x}$,then the equation of the curve is . . . . . . .