- For each differential equation below, do the following steps.
- Describe what each variable or function is measuring (if possible at this stage), and give correct units.
- Describe what the equation is saying. Use phrasing like “If such-and-such is big, than such-and-such grows faster.”
- Explain why the relationships from the previous bullet point makes sense in terms of the story or physical situation.
- Let [latex]T(t)[/latex] be the temperature of a cooling object in degrees Celsius, and let [latex]t[/latex] be measured in seconds. Newton’s law of cooling state that [latex]T'(t) = -k(T(t) - T_{\text{air}})[/latex]. Here [latex]T_{\text{air}}[/latex] is the ambient air temperature.
- Let [latex]H(t)[/latex] be the height of a mountain measured in meters over a long period of time ([latex]t[/latex] measured in millions of years). Suppose [latex]H(t)[/latex] satisfies the differential equation [latex]H'(t) = -k (H(t))^{1/3}[/latex].
- Let [latex]y(t)[/latex] be the fish population in a lake being harvested at rate [latex]H[/latex] fish per year. Suppose [latex]y(t)[/latex] satisfies the differential equation
[latex]y'(t) = k y(t) - (m + c y(t)) y(t) - H[/latex]. Here, [latex]k y(t)[/latex] represents the birth rate, [latex](m + cy(t)) y(t)[/latex] the natural death rate, and [latex]H[/latex] the harvest rate.
- Skim through the article “Campus drinking: an epidemiological model” by J. L. Manthey, A. Y. Aidoo & K. Y. Ward. You’re not going to understand the whole article — that’s okay! But let’s try to figure out bits and pieces of it.
Here is their first differential equation from secion 2 of the article.
$$
\frac{dN}{dt} = \eta – \eta N – \alpha N P + \beta S + \epsilon P
$$- What does the variables N, S, and P represent?
- In the first differential equation, what terms represent college students transitioning to drinking more? Which one represent college students transitioning to drinking less?
- What is a main conclusion of the article?