# 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.