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CHAP.  18]      GRAPHICAL  METHODS  FOR SOLVING DIFFERENTIAL  EQUATIONS               169




                  The  above  results are displayed  in Table  18-1: For comparison,  Table  18-1 also  contains  results for h = 0.05,
               h = 0.01, and h = 0.005, with all computations rounded to four  decimal  places.  Note that more accurate  results are
               obtained  when  smaller values of h are used.
                  If we plot (x n, y n)  for integer values of n between  0 and  10, inclusively, and then connect  successive points with
               straight  line  segments,  we  would  generate  a  graph  almost  indistinguishable from  Fig. 18-13,  because  graphical
               accuracy  with the chosen  scales  on the axes  is limited to one decimal  place.

         18.14.  Find y(0.5) for / = y; y(0) = 1, using Euler's method with h = 0.1.
                  For  this  problem, (x,  y)=y,  x 0 = 0,  and y 0=  1;  hence,  from  Eq.  (18.5),  y!,=f(x n,  y n)=y n.  With  h = 0.1,
                                f
               y(O.S)  = y s. Then, using Eq. (18.4) with n = 0,  1, 2, 3, 4 successively, we obtain
















                                                                          5
               Thus, X0.5) = y 5 = 1.610.  Note that since the true solution is  Y(x)  = if,  7(0.5) = e°-  = 1.649.


                                                 Table  18-1

                                Method:  EULER'S METHOD

                                Problem:  y' = y -x; y(0)  = 2
                                                y
                          x n                   n                    True  solution
                                 h = 0.1  h = 0.05  ft = 0.01  h = 0.005  Y(x)  = (f  + x + 1
                          0.0    2.0000   2.0000   2.0000  2.0000       2.0000

                          0.1    2.2000   2.2025   2.2046  2.2049       2.2052
                          0.2    2.4100   2.4155   2.4202  2.4208       2.4214

                          0.3    2.6310   2.6401   2.6478  2.6489       2.6499
                          0.4    2.8641   2.8775   2.8889  2.8903       2.8918

                          0.5    3.1105   3.1289  3.1446   3.1467       3.1487
                          0.6    3.3716   3.3959  3.4167   3.4194       3.4221

                          0.7    3.6487   3.6799  3.7068   3.7102       3.7138
                          0.8    3.9436   3.9829  4.0167   4.0211       4.0255

                          0.9    4.2579   4.3066  4.3486   4.3541       4.3596
                          1.0    4.5937   4.6533  4.7048   4.7115       4.7183
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