1.
What is a function? Find a formal definition of function (as a mathematical object), and then unpack that definition, explaining it in your own words. Make sure to attend to the important details!
Section 1.1 Modeling with Differential Equations
Exercises Exercises
Supplemental HW Problems.
2.
What is a solution to a differential equation? Be specific. Also explain why a solution is not the kind of algebraic solutions we’re used to (\(x = 3\text{,}\) for example) and why we need some of the mechanisms from exercise 1 of this exercise set.
3.
Consider the differential equation
\begin{equation*}
y'' + 4y = 0\text{.}
\end{equation*}
- Verify that \(y(t) = c_1 \cos(2t) + c_2 \sin(2t)\) is a solution to this equation.
- Sketch solution curve for \(c_1 = 1\) and \(c_2 = 1\text{.}\)
- If \(y(0) = 1\text{,}\) and \(y’(0) = -2\text{,}\) find \(c_1\) and \(c_2\text{.}\)
4.
The growth of a population of fresh-water Gars in a large lake (where resources and space are effectively unlimited) can be modeled by the differential equation
\begin{equation*}
\frac{dP}{dt} = kP\text{.}
\end{equation*}
- Unpack this equation in words. What do all of the different letters mean? What are their units?
- Classify the growth of the population for different values of \(k\text{.}\) Attend to values of \(k\) less than zero, zero, between zero and one, and greater than one.
- If we allow fishing of the Gars at a constant rate, what does that do to our equation – what differential equation can we use to model this?
- If we allow fishing of the Gars at a rate proportional to the population, what differential equation can we use to model this?
- Assume that \(N\) is the maximum number of Gars that the lake can contain (the carrying capacity of the lake). What differential equation can we use to model this? (HINT: think about what happens when \(P = N\text{,}\) \(P \gt N\text{,}\) \(P \lt N\)).
