Integration of Differential Equations. 599 



Thus if F"(x) = dy/dx is positive and 'dfl'by is also positive, 

 the result of section II., 



A<&<B, 

 may be replaced by 



p<k<Q, (7) 



where p = If (a + ^h, b + \mK) 



and Q = ih{f(a, b) + %f(a + \\ b + J>M/i) +/(a + 7i, 6 + M/t) } ; 



while if ¥"(x) is positive, and ~dfl~dy is negative, 



P<&<<7, («) 



where P = hf(a -f P, * + pi/ij 



and ? = £/*{/(«, ^)4 2/(« + i//, & + M)+/(a+/i, 6 + ?nA)j. 

 Similarly, if F"(^') and ~dffoy are both negative, 



p>£>Q (9) 



while if F" (.*.■) is negative and ~dfl~dy positive, 



V>k>q (10) 



These results may be summed up by saying that in every 

 case (subject to the limitations on /stated at the beginning 

 of this section) k lies between the greatest and least of the four 

 numbers p, P, q, and Q. 



As an approximate formula we use &=§B + JA, replacing 

 B by Q or q, and A by p or P. 



IV. Application to a numerical example. 



! examj 

 method i 



Consider the example selected by Runge and Kutta to 

 illustrate their methods 



1 oc 

 — ; w = l when x = 0. 



dx y + x J 



It is required to find the increment k of ;/ when x increases 

 by 0*2. Here f{x, y) = (y — x)l(y+x). This function satisfies 

 the conditions laid down in the last section. 



We take M = l, m = (l-02V(l-2 -+ 0'2) = 4/7. 



Then p=0'1654321, 



P = 0-1666667, 



2 = 01674987, 

 Q = 0-1690476. 



