§7.1 An introduction to differential equations (continued)

Solving a differential equation

Summary

$v(t) = 1.5t - 0.25t^2$.

$v(t) = 3 + 2e^{-0.5t}$.

$v(t) = 3$.

$v(t) = 3 + Ce^{-0.5t}$ where $C$ is any constant.

Suppose that $T(t)$ represents the temperature of a cup of coffee set out in a room, where $T$ is expressed in degrees Fahrenheit and $t$ in minutes. A physical principle known as Newton's Law of Cooling tells us that

. - Supposes that $T(0)=105$. What does the differential equation give us for the value of $\frac{dT}{dt}\vert_{T=105}$? Explain in a complete sentence the meaning of these two facts. - Is $T$ increasing or decreasing at $t=0$? - What is the approximate temperature at $t=1$? - On the graph below, make a plot of $dT/dt$ as a function of $T$. Coordinate axes for plotting $\frac{dT}{dt}$ as a function of $T$. The values of $T$ range horizontally from $T=0$ to $T=120$, and the values of $\frac{dT}{dt}$ range vertically from $\frac{dT}{dt}=-3$ to $\frac{dT}{dt}=5$- For which values of $T$ does $T$ increase? For which values of $T$ does $T$ decrease? - What do you think is the temperature of the room? Explain your thinking. - Verify that $T(t) = 75 + 30e^{-t/15}$ is the solution to the differential equation with initial value $T(0) = 105$. What happens to this solution after a long time?

In this problem, we test further what it means for a function to be a solution to a given differential equation. - Consider the differential equation

. Determine whether the following functions are solutions to the given differential equation. -$y(t) = t + 1 + 2e^t$-$y(t) = t + 1$-$y(t) = t + 2$- When you weigh bananas in a scale at the grocery store, the height $h$ of the bananas is described by the differential equation

where $k$ is the spring constant, a constant that depends on the properties of the spring in the scale. After you put the bananas in the scale, you (cleverly) observe that the height of the bananas is given by $h(t) = 4\sin(3t)$. What is the value of the spring constant?

Suppose that the population of a particular species is described by the function $P(t)$, where $P$ is expressed in millions. Suppose further that the population's rate of change is governed by the differential equation

where $f(P)$ is the function graphed below.

• For which values of the population $P$ does the population increase? - For which values of the population $P$ does the population decrease? - If $P(0) = 3$, how will the population change in time? - If the initial population satisfies $0\lt P(0)\lt 1$, what will happen to the population after a very long time? - If the initial population satisfies $1\lt P(0)\lt 3$, what will happen to the population after a very long time? - If the initial population satisfies $3\lt P(0)$, what will happen to the population after a very long time? - This model for a population's growth is sometimes called “growth with a threshold.” Explain why this is an appropriate name.