Module 22  Differential Equations and Euler's Method  
Introduction  Lesson 1  Lesson 2  Lesson 3  SelfTest  
Lesson 22.2: Euler's Method  
In this lesson Euler's method is used to approximate the solution to an initialvalue problem. The method is based on linear approximations and uses a variation of the pointslope form of a linear equation: y_{1} = y_{0} + m(x_{1} – x_{0}). Linear Approximations
Suppose we want to solve a differential equation of the form
where m(x, y) represents the slope of the function y = f(x) at the point (x, y). In general, this means that the slope of the graph of Euler's Method Euler's method, which is described below, approximates the yvalues of the solution for a sequence of xvalues given by x_{1}, x_{2}, x_{3}, ..., etc. The method uses linear approximation to move from one point on the solution curve to the next. At each point, it uses the equation to compute the slope of the tangent line and then moves along the tangent line to the next point.
Basically, since (rise) = (slope) · (run), starting at (x_{0}, y_{0}) and moving along the tangent line to the point x = x_{1} changes the yvalue by The next value may be found by repeating the process. The value of y_{2} = f(x_{2}) can be estimated from the value of y_{1} = f(x_{1}) by the linear approximation After approximating f(x_{2}) the same method can be used to approximate f(x_{3}). The value of f(x_{n+1}) can be approximated by using previously found values.
Using Euler's Method In Lesson 22.1 you graphed an approximate solution to the logistic initialvalue problem
Euler's method may be used to approximate the values of the solution to this differential equation. For this problem m(x,y) = 0.037 · y · (25  y). First let x_{0} = 1, y_{0} = 1, and x_{1} = 1.1, then compute Next, find f(x_{1}) using the linear approximation equation. Notice that x_{1}  x_{0} = 0.1
f(x_{1}) = f(1.1)
f(x_{0}) + m(x_{0}, y_{0}) · (x_{1}  x_{0}) You can use the expression for m(x_{0}, y_{0}) in the linear approximation equation to find the value of y directly. Using one expression to find yvalues will make the computation easier when finding subsequent values.
Approximate y_{2} = f(1.2) using the estimated value found for y_{1} and the value of m(x_{1}, y_{1}). y(1.2) = y_{2} = f(x_{2}) 1.0888 + m(1.1, 1.0888) · 0.1 = 1.0888 + [0.037 · 1.0888 · (25  1.0888)] · 0.1 Then continue the process by letting x_{3} = 1.3, x_{4} = 1.4, ... to find y_{3}, y_{4}, .... The ANS Feature There is a short cut that will speed up the process for subsequent approximations. The result of the most recent calculation is stored in a variable called ANS which can be entered into a calculation by pressing [ANS]. [ANS] is above . This approach will also improve the accuracy somewhat, since it uses all 14 places, rather than the 4 we entered by hand. Because the most recent result represents y(1.2), the approximation for y(1.3) can be found by evaluating y(1.3) Ans + 0.037 · Ans · (25 – Ans)· 0.1
Ans has now been updated to 1.28956.
Pressing reexecutes the previous command using the new value stored in ANS and displays the result. When the result is displayed, ANS is updated for the next computation.
22.2.1 Approximate y(2) by pressing
five more times.
Using Euler's Method in TI83 Programs
Euler's method is used in the EULERG and EULERT programs. EULERG produces a visual solution to the differential equation stored in Y1. In Lesson 22.1, the solution to the logistic differential equation Displaying a Table of Values You can see the Euler method values with the program EULERT using the same differential equation.
Each time you press you see the coordinates for another point. These are the same values produced by Euler's method using the ANS feature earlier in this lesson.
EULERG also uses Euler's method to calculate x and y coordinates; however, EULERG plots the points instead of displaying the coordinates. 

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