NUMERICAL SOLUTION OF DIFFERENTIAL EQUATIONS 



First Trial y-Table 



237 



Since y = f it now follows from the first equations of di) and 10.7 (4) for n = i 

 that an approximate value of x z is 



= o.iooo 



+ - - .9776 + - .0149 H --- .0074 = .1986. 



With this value of x z it is found from the second of (8) that g z = .2901. Then 

 the first trial g- table constructed from the values of g at / = o, o.i, 0.2, is: 



First Trial g-Table 



Then the second equation of 10.7 (4) gives for n = i the more nearly correct 



value of y 2 , 



i T i i 1 



14. y z = .9925 + ^-.2901 + .1411 - .oo79j = .9705. 



This value of y z should replace the last entry in the first trial y-table. When 

 this is done it is found that Ai^ 2 = -.0220, A^ = .0145. Then the first equa- 

 tion of 10.7 (4) gives 



15. x 2 = .1000 + .9705 + - .0220 + .0145 = .1983. 



The computation is now well started although Xi, y\, x z , and y z are still subject 

 to slight errors. The values of Xi and yi can be corrected by applying 10.31 for 

 n = i. It is necessary first to compute a more nearly correct value of g 2 by using 

 the value of x~ given in (15). The result is g 2 = -.2896, Aig 2 = -.1406, 

 A 2 g 2 = +.0084. Then the second equation of 10.7 (4) gives 



16. 



y, = .9925 + 



-[-.2896 + -. 



O |_ 2 



1406 - .0084 



12 



= -975> 



agreeing with (14). This value of y 2 is therefore essentially correct. An applica- 

 tion of 10.31 then gives 

 17. Xi = .0000 + ^ .9705 + | .0220 - -| .0145 = .0997, 



