Question Type 1: Formulating a model for differential equation given a contextual description Exercises

Given the model $\dfrac{dX}{dt} = -\dfrac{k}{X}$, where $X(0) = 10\text{ kg}$ and $k = 3\text{ kg}^2/\text{day}$, determine the number of days until $X$ reaches zero.

Given that $X(0) = 5$ and $k = 2$, find the time $t$ when $X = 1$.

Solve the differential equation $\frac{dX}{dt} = -\frac{k}{X}$ subject to the initial condition $X(0) = X_0$.

Explain why the solution $X(t)=\sqrt{X_0^2-2kt}$ implies a finite extinction time and contrast this with exponential decay models.

Assume $\frac{\text{d}X}{\text{d}t} = -\frac{k}{X}$ and that $X(0) = X_0$. Derive the general time $t_r$ for $X(t_r) = rX_0$, where $0 < r < 1$.

The concentration of a pollutant, $P$, follows the differential equation $\frac{\text{d}P}{\text{d}t} = -\frac{k}{P}$ for $P > 0$, where $t$ is the time in days and $k$ is a positive constant.

Given that $P = 50$ when $t = 0$, and that after $8$ days the concentration is $10$:

Determine the value of $k$ and the extinction time $T$ such that $P(T) = 0$.

Given the solution $X(t) = \sqrt{X_0^2 - 2kt}$, find the time $T$ at which $X(T) = 0$.

A sample decays according to the differential equation $\frac{\text{d}X}{\text{d}t} = -\frac{k}{X}$, where $X$ is the amount of the sample at time $t$, $k$ is a positive constant, and $X_0$ is the initial amount at $t=0$.

Given that the half-life of the sample is $t_{1/2}=4$, express $k$ in terms of $X_0$.

A reactor’s mass $M$ follows the differential equation $\frac{\text{d}M}{\text{d}t} = -\frac{k}{M}$ where $k > 0$, and the initial mass is $M(0) = 100$.

Find the time required to reduce the mass $M$ to $20$, and comment on whether the model allows for negative mass.

Show that the time $t_{1/2}$ for $X(t)$ to decay to half its initial amount $X_0$ is $t_{1/2} = \frac{3X_0^2}{8k}$ if the model is $\frac{dX}{dt} = -\frac{k}{X}$.

Express the time $t$ as a function of $X$ for the decay model $\frac{dX}{dt} = -\frac{k}{X}$ with $X(0) = X_0$.

Write the differential equation that models the decay of a material $X(t)$ if its decay rate is inversely proportional to the current amount $X$.