CSC #4, Part 1

Modeling with Differential Equations

What should you do with this worksheet?

There is nothing to turn in from this worksheet. Instead, you should read what follows, carry out instructions when given, and in general experiment with the worksheet. When you understand the ideas contained in it, you will proceed to the second worksheet named CSC #4, Part2. The second worksheet contains problems and questions, similar to those presented in this worksheet, that you will investigate and discuss. You will then turn in a printed copy of a modified version of the second worksheet that contains your responses to the activities and questions.

What is this worksheet about?

The concept of derivative provides a powerful tool for studying functions. Given a function the derivative can reveal properties that are not so apparent from the definition or , for example where is increasing?, where is it decreasing?, where does it take on its maximum and minimum values? The second derivative measures even more subtle properties of the function - when does the rate of change of reach maximum or minimum values, i.e. where are the inflection points of ? All of these applications of derivatives involve us in the task ... given , find its derivative .

The converse problem is even more interesting - given the derivative of a function, find the function itself. For example, given the speed of an object find its location, given the federal deficit find the accumulated national debt, given the birthrate of a population predict the population of the country 50 years hence, given the transmission rate of a disease predict the long-term progress of an epidemic, given the acceleration of a planet determine its orbital characteristics. Such problems are described naturally in terms of differential equations - equations that involve an unknown function and its derivatives. The problem, then, becomes that of solving the differential equation for the unknown function.

You have already encountered differential equations in at least three contexts. In section 4.8 in your text you studied differential equations that can be written in the form

Then the general solution is the general antiderivative , where . The arbitrary constant in the general solution indicates that there is actually a family of solutions of the differential equation. A particular solution can be picked out by specifying a point ( , ) through which it should pass. As noted on page 288 in your text, such problems are called initial-value problems - given a differential equation together with an initial point on the desired solution curve, find the solution.

Example 1. Consider the initial-value problem , when . This problem is simple enough to solve directly by finding the general anti-derivative . Substituting the initial values into this general solution, we obtain , or . Thus we have found the particular solution .

Example 2: A second occasion on which you met a differential equation was in CSC #2. Here again an initial-value problem arose in the context of finding how much water flows from a cylindrical tank in time . The differential equation

was derived using physical principles such as Torricelli' s Law along with basic geometrical facts. Together with the initial condition , where was the depth of the water at time , we once again had an initial-value problem. This one was more complicated because both the unknown function as well as its derivative appear in the equation. You solved this equation using the method of "guessing and checking", and again later using the method of separation of variables .

Our objective in CSC #4 is to strengthen our ability to explore differential equations that arise in the context of real world problems. So far our methods of solution are meager. In much more advanced courses one learns to solve a wide variety of differential equations, but even then it is quite usual to encounter differential equations in applications that defy all the methods and tricks we have learned. In this case study we will add numerical methods to our arsenal of techniques for exploring differential equations.

Contents of the Following Sections:

• Activate me before using this worksheet - defines a utility function iterate .
• Getting Started - Interpreting and using slope fields
• Euler's Method - numerical and graphical solutions of differential equations.
• Modeling Population Growth - an application of Euler's method.
• Summary and Conclusions.
• Exercises

Activate me before using this worksheet.

Getting Started: Interpreting and Using Slope Fields

Euler's Method: Numerical and Graphical Solutions

Modeling Population Growth using Euler's Method

Summary and Conclusions

Program for Euler's Method

Exercises