# 5.6: Homework- Understanding Differential Equations

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1. 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.
1. 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.
2. 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 .
3. 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.
2. 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.

$\[\frac{dN}{dt} = \eta - \eta N - \alpha N P + \beta S + \epsilon P$\]

1. What does the variables N, S, and P represent?
2. In the first differential equation, what terms represent college students transitioning to drinking more? Which one represent college students transitioning to drinking less?
3. What is a main conclusion of the article?

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