Chapter 9 Lesson 34: Modeling with ODEs I

9.1 Objectives

  1. Given a description of a rate and an initial condition, identify an IVP that matches that description.

  2. Set up and/or solve an IVP that models the changing temperature of an object placed in a fixed environment using Newton’s law of heating/cooling.

  3. Set up and/or solve an IVP that models a growing population without a carrying capacity.

  4. Set up but do not solve an IVP that models a growing population with a carrying capacity.

  5. Set up an IVP that models the changing amount of an investment or debt account, given an interest rate and repeated investments or repeated payments

9.3 In class

  1. Review of ODEs. Start with a reminder of where we’ve been the last few lessons. We’ve worked with separable ODEs including pure-time and autonomous ODEs. We’ve found solutions to IVPs involving these using analytical methods, and approximated solutions using graphical and numerical methods. Today we’ll start talking about why we care about ODEs. It turns out that ODEs provide us with a language that clearly describes many natural phenomena. Many formulas which students are familiar with (projectile motion with no wind resistance, population growth being two) aren’t obvious at all. Just by looking at the formula, it isn’t clear that the formula describes anything at all. Those formulas are just solutions of ODEs, and if you know the language of ODEs it is clear that the ODE is describing something.

  2. Putting it Together. Now we will focus on using ODEs for the purpose of modeling. Some of this should be review from block 1, but we will discuss basic growth/decay problems, and then move to growth problems with carrying capacity using the logistic equation. Demonstrate using the squirrel example from the textbook.

  3. The book contains lots of material. This is partly due to Rob’s original material, and partly due to my own tendencies. Obviously you won’t be able to cover all of those models in one day. Focus on exponential growth (population and interest), logistic growth, and Newton’s Law of Cooling.

  4. Today is not about solving. It is about a) interpreting what the ODE says, so that students can set up similar IVPs when given information, and b) using the solution to an IVP to answer a question. You shouldn’t spend a lot of time solving these equations. They should, at this point, remember the solution to \(y'=ky\). The solution to \(T'=k(T_a-T)\) you can do quickly, or assign as an exercise. Don’t spend time solving logistic models. We didn’t do partial fractions decomposition in this class. Plus, that’s not the point. Writing down and interpreting the models is and using the solutions of the models to answer questions is the point.

  5. Examples Work through as many examples as you can. Blow over the solving of the ODEs. Talk about interpreting what the ODEs say and using the solutions of ODEs to answer questions.

9.4 R Commands

findZeros, antiD

9.5 Problems & Activities

  1. Start with exponential growth. Interpret the differential equation \(P'=kP\) as “the rate of growth of the population is always proportional to the current population”. That means that \(k\) must be of dimension time\(^{-1}\). Write down the general solution, \(P=P_0e^{kt}\), and remind them, from Block 1, of the relationship between \(k\) and doubling time. With the general solution in hand, work through a few of the questions in Example 8.1.2.1.

  2. Another nice example of exponential growth is continuously compounded interest. Interpret the differential equation \(P'=rP+I\) as “the rate at which the account value grows is the rate at which interest is accumulated, plus the rate at which money is deposited”. The rate at which interest is accumulated is modeled as \(rP\), where \(r\) is the interest rate. The solving of that ODE should be a forgone conclusion now. Solve if if you feel you have time, but feel free to write down the general solution, \(P=\left(P_0+\frac{I}{r}\right)e^{rt}-\frac{I}{r}\). Then, do Example 8.1.3.1. Note that using Euler’s method to solve the ODE numerically, and interpret the output of that is also totally fine.

  3. Now discuss a similar growth process, but note that in many cases, a quantity will realistically not be allowed to grow infinitely. At some point, a “carrying capacity” is reached, and there is simply no more room to grow. Consider plants or animals in a finite space. These problems are often modeled using a logistic equation: \[P^{'} = kP\left(1 - \frac{P}{M}\right)\], where \(k\) is the growth rate when the population is small and growing exponentially, and \(M\) is the carrying capacity. Don’t try to solve this ODE. Instead follow Example 8.2.0.1, and use Euler to do the solving work for you.

  4. Now, discuss Newton’s Law of Cooling. Again, solving this analytically in class isn’t the point, it should be do-able on homework. Feel free to either assert that the solution is \(T=(T0 - T_A)e^{-kt} +T_A\), or just skip completely! It is interesting how much information we can extract without solving the ODE analytically. Cover Example 8.4.1.

  5. As time remains, cover more examples. There are plenty in the book.