Lesson 20

Getting Started (Introduction to Differential Equations, continued)

We will begin this lesson by considering what a differential equation is and what a slope field represents.

The simplest definition of a differential equation is an equation that involves a derivative. In calculus, you will learn how to find the derivative of a function and what it means. A “derivative” is represented by a form similar to dy/dx or y’. A “differential equation” is an equation that contains derivatives.

The simplest definition of a derivative is that it is the rate of change (slope) of the function at a given x-value. For example, if the function is linear, the slope is a constant rate of change and so the derivative is constant. Given y=3x+1, we know that the slope is 3 and so y’=3.

If the function is not linear, the derivative is very useful! It helps us find the rate of change at any individual point on the function.

To draw a slope field, we can use dy/dx or y’ to find the slope of the original curve at a single x-value. We then draw a short segment about the x-value that has the estimated slope (dy/dx). If we do this enough times, we will be able to visualize the original curve!

Part I

In this part, you will learn how to graph a 1st order differential equation and try to understand what the graph represents.

1. Graphing a Slope Field for a 1st Order Differential Equation

2. Inputting an Initial Condition Point using the IC Page

Notice that the solution curves we draw are linear! Does this make sense? Hmmm…

y’=2 (a constant) which implies that the original equation is a linear equation with a slope of 2. The original must be of the form y=2x+b. But, we would need additional information to find the exact value of b.

We could find b for (2,-3) and (1, 3).
For (2,-3) y=2x+b becomes -3=2*2+b -> b=-3-4=-7
What will b equal for (1, 3)? Guess by looking at the above graph!

3. Graphing Solution Curves by Guessing an Equation

Part I


Practice Exercises

Before beginning the practice exercises, open a word document, type in the following information and then save it as Lesson20 in your CASIO folder within My Documents:

  1. Please begin by opening the DiffEqGraph application.
  2. Click the Graph tab and select Edit/Clear All.
  3. Click the IC tab and select Edit/Clear All.
  4. Click the DiffEq tab and select Edit/Clear All.
  5. Following y', input 1/y and press EXE.
  6. Display the graph.
  7. With your equation and graph showing, get a screen capture. Paste it into your Lesson20 document (under a title of PART I).
  8. Click the IC tab and input x=1, y=2 and x=1, y=-2 for initial conditions.
  9. Display the graph with solution curves.
  10. With your equation and graph showing, get a screen capture. Add two blank spaces following the first screen capture and then paste this one.
  11. Click the Graphs tab. Following y, input (2x)^(1/2) and press EXE.
  12. Place focus on the line containing (2x)^(1/2) and change the line style to be thicker.
  13. Display the Graph.
  14. With your equation and graph showing, get a screen capture. Add two blank spaces following the last screen capture and then paste this one.
  15. Please clear the DiffEq page, IC page and the Graphs page. Each needs to be done separately.