The population $p$ of the city at time $t$ is given by $\frac{dp}{dt} = \frac{p}{2} - 100$. If the initial population at $t = 0$ is $100$,then $p$ is:

The differential equation $x \frac{dy}{dx} = 2y$ represents ....

The slope of a curve passing through the point $(2, 0)$ at $(x, y)$ is $x^2 - 2x$. Find the value of $y$ at $x = 0$.

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

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:

$A$ radioactive sample has a half-life period of $10 \text{ days}$. If a sample originally has a mass of $1000 \text{ mg}$,then the mass remaining after $50 \text{ days}$ is: