600 Numerical Integration of Differential Equations. 

 Thus k lies between p and Q. Errors. 



|Q+i 7 , = 0-1678424, 0*0000007 



. Kutta's value 0-1678449, 0*0000032 



Runge's value 0'lfi78487, 0*0000070 



Henri's value 0*1680250, 0*0001833 



The second, third, and fourth of these were calculated by 

 Kutta. They are also given in Bateman's * Differential 

 Equations,' p. 22i). Now this particular example admits of 

 integration in finite terms, giving 



log(^ + 2/ 2 )-2tan^(.%)=0. 



Hence we may find the accurate value of k. 



Accurate value = 0-1678417. 



Thus in this example our result is the nearest to the 

 accurate value, the errors being as stated above. 



We may also test the method by taking a larger interval 

 A = l. Of course a more accurate way of obtaining the 

 result would be to take several steps, say 76 = 0*2, 0*3, and 

 finally 0*5, as Runge does. 



Still, it is interesting to see how far wrong the results 

 come for the larger interval. 



We take M = l, m =(l-l)/(2 + 1) =0. 



Then tQ +.»/) = 0*50000. 



True value = 0-49828, Errors. 



Kutta's value = 0-49914, 0'00086 



Our value =-0-50000, 0'00172 



Heun's value = 0-51613, 0'01785 



Runge's value = 0-52381, 0'02553 



This time Kutta's value is the nearest, and ours is second. 



University College, Nottingham, 

 February 22nd, 1919. 



