5.6: Homework- Understanding Differential Equations
-
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 \(T(t)\) be the temperature of a cooling object in degrees Celsius, and let \(t\) be measured in seconds. Newton’s law of cooling state that \(T'(t) = -k(T(t) - T_{\text{air}})\). Here \(T_{\text{air}}\) is the ambient air temperature.
- Let \(H(t)\) be the height of a mountain measured in meters over a long period of time (\(t\) measured in millions of years). Suppose \(H(t)\) satisfies the differential equation .
-
Let \(y(t)\) be the fish population in a lake being harvested at rate \(H\) fish per year. Suppose \(y(t)\) satisfies the differential equation
\(y'(t) = k y(t) - (m + c y(t)) y(t) - H\). Here, \(k y(t)\) represents the birth rate, \((m + cy(t)) y(t)\) the natural death rate, and \(H\) 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?