Solving First Order ODE's

Rick Downs, South Seattle College, Mathematics

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Summary

This assignment is for a Differential Equations course that has students use a computer to solve first order ODE's graphically using slope fields, numerically, and analytically. One part of the assignment has students set up and solving a mixing problem for a one-compartment model of chemicals being dumped into a lake.

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Learning Goals

The goals of this assignment are for students to learn
- how to use a computer to solve first-order ODE's;
- create computer generated slope fields and identify equilibrium solutions;
- to numerically solve first-order ODE's using Euler's Method and the MATLAB ode45 command;
- to analytically solve first order ODEs using the MATLAB dsolve command; and
- set up and solve a mixing problem using MATLAB.

Context for Use

The assignment is designed to be used in an introductory Differential Equations course. This is an individual assignment to be completed after students have completed the MATLAB Onramp tutorial and students have learned how to solve first-order differential equations. The assignment should take about 2 weeks to complete. The assignment is used in a class that typically has 20 students.

Description and Teaching Materials

The materials provided are listed below:
- Project instructions for Students
- Rubric
- Project Solutions – Instructor Access Only
Project 1 - Solving First Order ODEs.docx (Microsoft Word 2007 (.docx) 60kB Nov9 21)
Project 1 - Rubric.docx (Microsoft Word 2007 (.docx) 17kB Nov8 21)

Teaching Notes and Tips

Before starting on this assignment, students should have:

  • Completed the MATLAB Onramp tutorial.
  • Familiarity with slope fields, equilibrium solutions, solving first-order ODE's, and Initial Value Problems.
  • Practice setting up mixing problems.
  • Experience working with Euler's Method.

Teaching Tips

  • It was necessary to be explicit about what to turn in, because sometimes students will submit a graph but no code, or the code but no graph. It is also necessary to ask students to state or explain the results from MATLAB. Many students think providing a copy of MATLAB's output is all that is needed for a solution.
  • Some students have different answers for questions 1a and 1b, 1c, 1d, without thinking that the answers should agree.  
  • Students new to programming may have trouble implementing Euler's Method in Problem 3. 
  • Some students do not choose an appropriate scale for the x- and y-axis. 

Assessment

A rubric is provided for students to let them know what they need to turn in for each part of the assignment. The solution to each problem is provided for instructors along with suggestions on the distribution of points for different parts of each problem.

References and Resources

MATLAB Onramp - a tutorial for students with no experience using MATLAB.
https://www.mathworks.com/learn/tutorials/matlab-onramp.html