232 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS 



10.7 The Step-by-Step Construction of the Solution. Suppose the differential 

 equations are 



with the initial conditions x = a, y = b at / = o. It is more difficult to start a 

 solution than it is to continue one after the first few steps have been made. There- 

 fore, it will be supposed in this section that the solution is well under way, and 

 it will be shown how to continue it. Then the method of starting a solution will 

 be explained in the next section, and the whole process will be illustrated 

 numerically in the following one. 



Suppose the values of x and y have been found for / = /i, / 2 , . . . . , t n . Let 

 them be respectively Xi, y\\ #, 3V, . . .; x nt y n , care bejng taken not to confuse 

 the subscripts with those used in section 10.6 in a different sense. Suppose the 

 intervals / 2 - t\, t* t z , ...,/ / n _i are all equal to h and that it is desired 

 to find the values of x and y at /+!, where t n +\ t n = h. 



It follows from this notation and equations (2) of 10.6 that the desired 

 quantities are 



// + ! 



-x n + f y(%?jd&G 



* 1 +I 



3>n+i = y n + / g(x, y, t) dt. 



^J f 



The values of x and y in the integrands are of course unknown. They can be 

 found by successive approximations, and if the interval is short, as is supposed, 

 the necessary approximations will be few in number. 



A fortunate circumstance makes it possible to reduce the number of approxi- 

 mations. The values of x and y are known at/ = /, /_!, / n _ 2 , . . . From these 

 values it is possible to determine in advance, by extrapolation, very close approxi- 

 mations to x and y for / = t n+l . The corresponding values of / and g can be 

 computed because these functions are given in terms of x, y, and /. They are 



also given for / = / B , tn-i, Consequently, curves for / and g agreeing 



with their values at / = t n+l , /, t n -i, .... can be constructed and the integrals 

 (2) can be computed by the methods of 10.1 and 10.3. 



The method of extrapolating values of x n+l and y n+ i must be given. Since 

 the method is the same for both, consider only the former. Since, by hypothesis, 

 * is known for t = t n , *_i, /*_, .... the values of *, A^ n , A 2 * n , and 

 A 3 # n are known. If the interval h is not too large the value of A 3 #n+i is very 

 nearly equal to A 3 # n . As an approximation A 3 # n+ i may be taken equal to A 3 # n , 

 or perhaps a closer value may be determined from the way the third differences 



