236 MATHEMATICAL FORMULAE AND ELLIPTIC FUNCTIONS 



Suppose K 2 = - and let y = -r" Then equation (i) is equivalent to the 

 two equations 



7< 



which are of the form 10.50 (i), where 



and x = o, y = i at / = o. 



The first step is to determine the interval which is to be used in the start of 

 the solution. No general rule can be given. The larger / and g the smaller 

 must the interval be taken. A fairly good rule is in general to take h so small 

 that hfo and hgo shall not be greater than 1000 times the permissible error in the 

 results. In the present instance we may take h = o.i. 



First approximations to x and y at / = o.i are found from the initial conditions 

 and equations 10.8 (2) to be 



= o H i = o.iooo, 



= i H o = i.oooo. 



10 



It follows from (8) and these values of Xi and y t that 



( f(x^ l \ y^\ /i) = i.oooo, 





I 



:vi (1) , /O = -0.1490. 



Hence the more nearly correct values of Xi and y it which are given by 10.8 (3), are 





= o H [i.oooo + i.oooo] = o.iooo, 



II. 



i 0-I r -i 



= i H |_ '00o - 0.1490] = 0.9925. 



Since in this particular problem x = fy dt, it is not necessary to compute 

 both / and g by the exact process explained in section 10.8, for after y has been 

 determined x is given by the integral. It follows from (7), (8), (10), and (n) 

 that a first approximation to the value of y at / = / 2 = 0.2 is 



12. y^ = .0025 - -^ .1490 = .9776. 



With the values of y at o, .1, .2 given by the initial conditions and in equations 

 (9) and (12), the first trial y- table is constructed as follows: 



