### CA III.5 Differential Equations - Analytical Solutions

Purpose: To introduce differential equations in terms of both the dependent variable and the independent variable (and as a result exposing students to antiderivatives).

Classroom Procedure: Students should be given a few minutes to read through and try the first question and then it can be discussed as a class. Students can then be given more time to read and work on Question 2 and 3 - again with regular class discussion as needed depending on where groups are getting stuck. Question 4 should be started in class and completed outside of class.

Ideas this Activity Builds On: This activity builds on students understanding of the basic derivatives - including the different ways of expressing the derivative of the exponential function (in terms of the dependent variable and independent variable as discussed in Module II).

Introduction/Motivation of the Activity: To use our knowledge of derivatives to understand the form of a changing quantity from information about its derivative  - allowing us to use this ability to answer application questions.

Need to Establish by the End of Activity/Wrap-Up: What it means to be an analytical solution to a differential equations. Which types of differential equations have we been successful in finding analytical solutions for. When we have a differential equation with respect to the independent variable finding a solution is the same as finding an antiderivative. Why are differential equations with respect to the independent variable different to differential equations with respect to the dependent variable.

Additional Notes: There are no synthesis questions for this activity. In many ways the next activity serves as the synthesis for this activity as we move to numerical solutions for differential equations.

#### Teaching Journal: CA III.5

• Untitled Post  The main problem students have with this activity is realizing the difference between the differential equations with respect to the dependent variable and the independent variable. Other than that they work pretty well through this. The discussion of this difference usually comes up in Question 3 where they are forced for the first time to work with a differential equation that is with respect to the independent variable. It is probably worth collecting and grading Q4 or discussing it in class as students often still struggle with these questions and the next time they see them will probably be on the Module III exam.
Posted Jun 13, 2012, 6:49 PM by Mairead Greene
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