Differential Equations

A differential equation is a type of equation in which at least one of the variables is a derivative. Due to the nature of differential equations and the integrals used in solving them, only a fraction of differential equations are solvable, and only a fraction of those are solvable by the methods available in the scope of the AP Calculus BC course. One very common application of the differential equation is for modeling exponential growth and decay.

Algebraic Approach

As previously mentioned, not all differential equations are solvable. The only available method for solving differential equations within the scope of our course is by isolating variables. That is, treat the derivative as the fraction dy/dx, and isolate all x terms with dx and all y terms with dy and integrating. Then simplify the equation and solve for constant values given any variable values that you are given.


Common Mistakes

Common mistakes include forgetting or misusing the constant of integration. Although you may be integrating both sides of the equation, the constant of integration is only needed on one side as the difference of two constants of integration. Do not forget to treat it as a constant and rename as needed as you modify it throughout the solving of a problem.

Graphic Approach

Slope Fields

When given a differential equation, in which the derivative can be stated as an independent variable, a slope field can come in quite handy. A slope field is a plane in which there are a large number of short line segments, each centered on a point on the graph. Only a certain number of points are tested in creating a slope field, and at each point a line segment is created with a slope equal to the derivative at that point. The more points along each axis in a given window tested for the slope field, the more accurately it can depict a differential equation. However, since each point requires its own calculation, a huge number of points would simply lead to loads of tedious arithmetic. Once a slope field is made, you can pick a point to start from, and just eyeball what the graph would look like.


Euler's Formula

Euler, the famous mathematician, had a nice idea about how to come up with a more accurate graph from a slope field. Instead of using all the slopes from the slope field that wont be anywhere near your graph, use a larger number of values applicable to the desired graph. How might this be accomplished? Start by picking a starting point, and a value for dx. From the derivative at that point and the decided value for dx, you can determine a value for dy. From there you can generate another point on your graph by adding dy to your initial y value, and dx to your initial x value. Granted, this is still an estimate and not entire accurate. By assigning a value to dx, you are using a delta-x as an approximation for dx, which is truly the limit of delta-x as delta-x approaches 0. So, the smaller the dx you pick, the more accurate your results will be. However, to get an appreciable graph from a very small dx, you will need to repeat the process a huge number of times. Euler didn't have access to wonderful things like the computer you are reading this on, but you do. Unlike a person, a computer is capable of repeating a simple but time consuming process over and over again extremely quickly, and that is exactly what Euler's Formula is. While newer calculators with Computer Algebraic Systems may have differential graphing built in, you will need a special program to do it yourself, and pick your own values for things like dx.


Because solving a differential equation with a large or precise slope field or using Euler's Formula with a large number of points can be extremely calculation intensive, we're going to use a graphing calculator. Below are step by step instructions on how to use a program for a TI-83/4 as well as a similar program for a TI-89. TI-Connect is a free program provided by Texas Instruments with which you can download programs into your calculator. With TI-Connect installed, save the files on your computer, plug your calculator into your computer, right click the file where you saved it, and select "Send to TI Device." Use the new window that opens to execute the transfer into your calculator.

For a TI-83 or TI-84 you will need to download 2 programs. The first, EULERF, is the only one you will need to use, the second, SLOPE, simply contains data that the first calls on to run properly.

To use the program, you must have a differential equation in which the derivative is the independent variable. Put your expression for the derivative in the Y1= menu, and then run the program.

At the menu, you can select whether you want to generate a set of points as a solution or graph a slope field. Start by selecting SOLUTION.

The program will then prompt you for a starting coordinate for your solution, as well as the fixed value of dx.

At the TYPE menu, you have an option. STEP allows generate points one at a time, whereas INSTANT will generate a number of points instantly.

If you selected STEP, it will start by showing you your starting coordinates, and then each time you press enter, it will generate another point. When you are done, press the ON button to break the loop and end the program.

If you selected INSTANT, it will prompt you for the number of points you want to generate, and then find them all in rapid succession.

Regardless of which type of solution you pick, the set of coordinates will be saved in the Lists L1 and L2. After running the solution program, set up a stat plot for L1 and L2, and run the program again. This time select SLOPE FIELD at the main menu, and it will give you your final product.

For solving problems with a TI-89 Calculator, you will only need a single file, since it can build a slope field without an extra program. Unlike an 83 or 84, the TI-89 can solve for differential equations using Euler's method. However, the built in differential equation grapher does not allow you, the user, to select the fixed value for dx. The program, eulerf, works just like the one for the 83 or 84, except it doesn't do slope fields, and is coded for a TI-89.

Unlike the 83/84 version, you also do not need to store your equation in the y'= page for the program to make a solution, although you will need to for the slope field. Select the program from the User-Defined Catalog to begin.

Here, it will prompt you for the expression for your derivative. Simply type it in with the x and y variables.

It will then prompt you for initial and fixed values.

Select Step or Instant from the drop down menu. Step generates one point at a time, while Instant finds a number of points all at once.

If you selected Step, just press enter to generate each point, and press On to break the loop and quit the program.

If you selected Instant, enter the number of points you want, and it will find them.

After running the program, all of your generated values will be in list1 and list2. Set them up in a stat plot, put your equation on the y'= menu on Differential graphing mode, and view.

Practice Problems

Algebraic Problem

Solve this differential equation for y. Assume there is a point at (0,2)

Slope Field Problem

Draw a slope field for this differential equation for all integer values of x and y between -2 and 2, and draw a possible solution for it starting at the point (1,1)

Euler's Formula

Use your calculator to draw slope fields and solutions to both the Algebraic Problem derivative and the Slope Field Problem derivative.

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