NUMERICAL SOLUTION OF DIFFERENTIAL EQUATIONS 



2 39 



first steps are very simple and can be carried out in practice in a few minutes if 

 the chosen time-interval is not too great. 



The problem now reduces to simple routine. There are an stable, a y-table 

 (which in this problem serves also as an /-table), a g-table, and a schedule for 

 computing g. It is advisable to use large sheets so that all the computations 

 except the schedule for computing g can be kept side by side on the same sheet. 

 The process consists of six steps: (i) Extrapolate a value of g n+l and its 

 differences in the g-table; (2) compute y n +i by the second equation of 10.7 (4); 

 (3) enter the result in the y- table and write down the differences; (4) use these 

 results to compute x n +\ by the first equation of 10.7 (4) ; (5) -with this value of 

 x n +i compute g n +i by the g-computation schedule; and (6) correct the extrapolated 

 value of gn+i in the g-table. 



Usually the correction to g n +i will not be great enough to require a sensible 

 correction to y n+ i. But if a correction is required, it should, of course, be made. 

 It follows from the integration formulas 10.7 (4) and the way that the difference 

 functions are formed that an error e in gn+i produces the error f he in y^, and 



the corresponding error in x n +i is -j- h 2 e. It is never advisable to use so large 



04 



a value of h that the error in x n +i is appreciable. On the other hand, if the differ- 

 ences in the g-table and the y-table become so small that the second differences 

 are insensible the interval may be doubled. 



The following tables show the results of the computations in this problem 

 reduced from five to four places. 



Final #-Table 



