Euler's Method

By generating slope fields, you can estimate solution curves by determining the tangent at various points in the plane. If you take this concept one step further, you can determine an actual solution curve numerically. This method is called Euler's method. This geometrical approach to solutions is very revealing because of its simplicity. Although not very accurate, Euler's method is studied because of this simplicity and because it is basic to the understanding of other more sophisticated numerical methods.

LiveMath has the capability of generating tables of solutions using this method, and after you look at how the method is derived, this feature of the program will be described.

The key to understanding Euler's method, as was the case with slope fields, is the fact that a differential equation gives you slope values for given inputs. If given an initial value () to a differential equation, you know that the point lies on the solution curve. Using the example dy/dx = y, you also know that the slope at any point is the y value. By drawing a straight line representing that slope at an initial you will have a curve which approximates the actual solution curve. It is a linear approximation which becomes less and less accurate the farther away from that you get.

If you choose a second x, some distance away, say x + x, you can determine a new y value by applying some simple algebra. After determining that second point, you can generate a new slope value using the differential equation. A second line is then drawn with this new slope and you proceed on to a third point. This process is repeated for the domain of the problem.

Using the same equation (dy/dx = y), the following graphic demonstrates this process for the first point.

The graph below is not to scale. The steps are labeled 1 thru 5.

If you look at #5 in the graphic, you will see that the new y value is now equal to 1.10. Since the x value is given as .10, you can repeat the process again to determine a new point at x =.20. The first four points are shown in the second graph.

Since you know the solution to this equation, generate a table in LiveMath and plot it with the actual solution curve in the same graph theory.

Expl #20 Euler vs Actual Solution

Input the following two Props:

You can construct the derivative as shown, or you can use LiveMath's partial derivative Op.

Start by selecting the equation (click on the equal sign) and choose Integrate Differential Equation..., under the Compute menu. The first thing that you need to do is to select Euler rather than Runge-Kutta by clicking on the latter and choosing the former when the ODE Solver opens.

Input the information shown above and click OK. The table will be generated with 21 points according to Euler's method. Select the table by clicking on its leading icon and plot the points by choosing Linear under the Graph menu. The table and the resulting graph theory (zoomed out) are shown below.

You know that the solution curve is represented by the equation , so duplicate the first line plot inside the graph details (by selecting and Copy/Paste). Replace the y in the line description with .

Change the color of the line. Before this plot will show up, you must give C must a value. This value is equal to 1. Do this and the plot will be generated.
Next, to better display the Euler line, generate a scatter plot of the points. Select the table again and choose Add Scatter Plot under the Graph menu. LiveMath will automatically add the 21 scatter points over the Euler line. The resulting graph theory is displayed below.

The Euler curve is a pretty good approximation close to the starting point, but gets worse the farther out you go. To analyze the actual error the following case theory is offered.

Copy the original table and paste inside a case theory. Rename the table E (User Defined Function). Next define C=1 and x=0.1. Set up two equations, the first defining the actual solution and the second as the table E(x). Call this second y,euler. Finally, create an error equation defined as the difference. Calculate this equation to determine the error. Now you can change the value of x to see different errors for different values of x. Click here to see a Livemath demonstration.

You can study the errors further by generating another table.

Select the Error equation and generate a table with the same data used to generate the original Euler table. This is displayed below along with a graph of the errors. To generate the error graph, select the table and choose Linear under the Graph menu. Make x the x-axis and Error the y-axis. Click here to see a Livemath demonstration.

As you can clearly see, the Euler approximation is not accurate except for values very close to the starting point. The problem is the fact that the Euler method uses the slope of the left hand end point of each interval and assumes that it reflects the slope over the entire interval. The size of the error can be lessened by choosing a smaller interval ( x), but this can cause other problems, mostly due to roundoff errors.

A second method, commonly referred to as the "Improved Euler Method", attempts to solve this problem by taking the slope values at each end of the interval and averaging the two. This method will not be discussed, deferring instead to a much more accurate and more practical technique called the Runge-Kutta method.