2 34 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS 



10.8 The Start of the Construction of the Solution. Suppose the differential 

 equations are again 



with the initial conditions x = a, y = b at t = o. Only the initial values of x and 

 y are known. But it follows from (i) that the rates of change of x and y at t = o 

 are f(a, b, o) and g(a,b, o) respectively. Consequently, first approximations to 

 values of x and y at t = h = h are 



2 - 



*, M>). 



Now it follows from (i) that the rates of change of x and ? at x = x ly y = y lt 

 f = t, are approximately f(xF>, yi (1 \ /O and g(xP\ y^\ ti). These rates will be 

 different from those at the beginning, and the average rates of change for the 

 first interval will be nearly the average of the rates at the beginning and at the 

 end of the interval. Therefore closer approximations than those given in (2) to 

 the values of x and y at / = ti are 



a, b, o) 

 (a, b, o) 



The process could be repeated on the first interval, but it is not advisable when 

 the interval is taken as short as it should be. 



The rates of change at the beginning of the second interval are approximately 

 f(xi (2 \ yi ( *\ ti) and g(xi (2) , yP\ ti) respectively. Consequently, first approxima- 

 tions to the values of x and y at t = t 2 , where t 2 h = h, are 



With these values of x and y approximate values of / 2 and g 2 are computed. Since 

 /o, go; fiy g\ are known, it follows that Ai/ 2 , Aigy, A 2 / 2 , and A 2 g 2 are also known. 

 Hence equations (4) of 10.7, for n + i = 2, can be used, with the exception of 

 the last terms in the right members, for the computation of x 2 and y z . 



At this stage of work x = a, y = 6; Xi, y\\ x z , y 2 are known, the first pair 

 exactly and the last two pairs with considerable approximation. After / 2 and g 2 

 have been computed, Xi and yi can be corrected by 10.31 for n = i. Then ap- 

 proximate values of x 3 and y 3 can be extrapolated by the method explained in 

 the preceding section, after which approximate values of / 3 and g z can be com- 

 puted. With these values and the corresponding difference functions, x 2 and y 2 

 can be corrected by using 10.31. Then after correcting all the corresponding 

 differences of all the functions, the solution is fully started and proceeds by the 

 method given in the preceding section. 



10.9 Numerical Illustration. In this section a numerical problem will be treated 

 which will illustrate both the steps which must be taken and also the method of 



