NUMERICAL SOLUTION OF DIFFERENTIAL EQUATIONS 235 



arranging the work. A convenient arrangement of the computation which pre- 

 serves a complete record of all the numerical work is very important. 

 Suppose the differential equation is 



ffi 



dx 



o, = i at t o. 

 at 



The problem of the motion of a simple pendulum takes this form when expressed 

 in suitable variables. This problem is chosen here because it has an actual physi- 

 cal interpretation, because it can be integrated otherwise so as to express / in 

 terms of x, and because it will illustrate sufficiently the processes which have 

 been explained. 



Equation (i) will first be integrated so as to express t in terms of x. 



On multiplying both sides of (i) by 2 and integrating, it is found that the 

 integral which satisfies the initial conditions is 



2. 



On separating the variables this equation gives 



dx 



(i - x 2 ) (i - 

 Suppose K 2 < i and that the upper limit x does not exceed unity. Then 



4. ~J= == = I + - K 2 * 2 + g * 4 * 4 + , 



Vi - K 2 x* 8 16 



where the right member is a converging series. On substituting (4) into (3) and 

 integrating, it is found that 



5. t = sin" 1 x + iC-~ x Y / r x 2 + sin" 1 x~\K 2 + f [_ x*^/i x 2 f#(i s 2 )* 



+ faA/i - x 2 + f sin- 1 x~] K 4 + ]. 



When x = i this integral becomes 



6. T = 



Equation (5) gives / for any value of x between -i and +i. But the problem 

 is to determine x in terms of t. Of course, if a table is constructed giving t for 

 many values of x, it may be used inversely to obtain the value of x corresponding 

 to any value of t. The labor involved is very great. When K 2 is given numerically 

 it is simpler to compute the integral (3) by the method of 10.1 or 10.3. 



In mathematical terms, / is an elliptical integral of x of the first kind, and the 

 inverse function, that is, x as a function of /, is the sine-amplitude function, which 

 has the real period $T. 



