[size=100]If you do not need to find an exact solution to the initial value problem [math]y'=f(x,y)[/math] for the initial value [math](x_0,y_0)[/math], you can use [b]Euler's method[/b] to find a numerical solution. The interactive figure illustrates this method. You can change the initial value [math]A[/math] by dragging it. You can change the length of the step ([math]dx[/math]) by sliding the slider bar. The blue graph is the solution curve. [br][/size]

In the spreadsheet, the notation x(A) is the value of the the [math]x[/math]-coordinate of point A. The notation f(A2) means to evaluate function [math]f[/math] at the value in cell A2. Function [math]f[/math] is the (exact) solution of the differential equation.[br][br]Notice the formula in cell B3. The formula states the current y-value (B3) is equal to the previous y-value (B2) increased by how fast the y-value is changing in that time period (the derivative 1 – y becomes 1 – B2) times the length of the step (dx).

[i]Developed for use with Thomas' Calculus, published by Pearson.[/i]