Page 182 - Schaum's Outline of Differential Equations

CHAP. 18] GRAPHICAL METHODS FOR SOLVING DIFFERENTIAL EQUATIONS 165

Fig. 18-12). The value l(x n+1) is taken to be y n+1. Thus

and

Hence, y n+i =y n + hy' n, which is Euler's method.

18.11. Give an analytic derivation of Euler's method.
Let Y(x) represent the true solution. Then, using the definition of the derivative, we have

If A.X is small, then

Setting Ax = h and solving for Y(x n + Ax) = Y(x n+1), we obtain

Finally, if we use y n and y' n to approximate Y(x n) and Y'(x n), respectively, the right side of (_/) can be used to
approximate Y(x n+i). Thus,

which is Euler's method.

Fig. 18-11