NUMERICAL SOLUTION OF DIFFERENTIAL EQUATIONS 231 



f(x, y, t) dt, 

 y o 



2. 



I 



I/O 



lg(x,y,t)dt. 



*Jo 



If x and y do not change rapidly in numerical value, then/(#, y, t) and g(x, y, f) 

 will not in general change rapidly, and a first approximation to the values of x 

 and y satisfying equations (2) is 



(*! = <* + f/(a,M)<&, 



Jo 



3' If 



y. = 6 + / (, *, dt, 



{ Jo 



at least for values of / near zero. Since a and b are constants, the integrands in 

 (3) are known and the integrals can be computed. If the primitives can not be 

 found the integrals can be computed by the methods of 10.1 or 10.3. 



After a first approximation has been found a second approximation is given by 





f/Oi 



Jo 



dt, 



dt. 



The integrands are again known functions of t because x\ and yi were determined 

 as functions of t by equations (3). Consequently x 2 and y 2 can be computed. 

 The process can evidently be repeated as many times as is desired. The nth 

 approximation is 



( C* 



\x n = a+ I f(x-i, yn-i, t) dt, 



t/o 



5- r 



\yn = b + / gOn-i, y*-i, t} dt. 



V. i/o 



There is no difficulty in carrying out the process, but the question arises whether 

 it converges to the solution. The answer, first established by Picard, is that, 

 as n increases, x n and y n tend toward the solution for all values of / for which all 

 the approximations belong to those values of x, y, and / for which / and g have 

 the properties of continuity with respect to t and differentiability with respect 



to x and y. If, for example, / = ^^ and the value of x n tends towards zero 



for t = T, then the solution can not be extended beyond / = T. 



It is found in practice that the longer the interval over which the integration 

 is extended in the successive approximations, the greater the number of approxi- 

 mations which must be made in order to obtain a given degree of accuracy. In 

 fact, it is preferable to take first a relatively short interval and to find the solution 

 over this interval with the required accuracy, and then to continue from the end 

 values of this interval over a new interval. This is what is done in actual work. 

 The details of the most convenient methods of doing it will be explained in the 

 succeeding sections. 



