NUMERICAL SOLUTION OF DIFFERENTIAL EQUATIONS 



223 



yi - y Q , 

 Ai}> 2 = yz- yi, 



= y n - 



These are the first differences of the values of the function y for successive values 

 of t. All the successive intervals for / are supposed to be equal. 

 10.21 In a similar way the second differences are defined by 



A 2 ;y 3 = 



10.22 In a similar way third differences are defined by 



and obviously the process can be repeated as many times as may be desired. 

 10.23 The table of successive differences can be formed conveniently from the 

 tabular values of the function and can be arranged in a table as follows : 



TABLE I 



A 2 ;y 



A 3 ;y 



y* 



In this table the numbers in each column are subtracted from those 

 immediately below them and the remainders are placed in the next column to 

 the right on the same line as the minuends. Variations from this precise arrange- 

 ment could be, and indeed often have been, adopted. 



10.24 A very important advantage of a table of differences is that it is almost 

 sure to reveal any errors that may have been committed in computing the y,-. 

 If a single y^ has an error e, it follows from 10.20 that the first difference AI>>; 

 will contain the error +e and Ai;y,-+i will contain the error e. But the second 

 differences A 2 ^ t -, A 2 ^t+i, and A 2 ^ t -+ 2 will contain the respective errors +e, 2, 

 +e. Similarly, the third differences AS?,-, As^t+i, A 3 ;yt+ 2 , and A 3 ;yt+3 will contain 

 the respective errors +e, -36, +36, -e. An error in a single y t affects j + i 

 differences of order j, and the coefficients of the error are the binomial coeffi- 

 cients with alternating signs. The algebraic sums of the errors in the affected 



