NUMERICAL SOLUTION OF DIFFERENTIAL EQUATIONS 227 



10.34 Now consider the application of 10.30 (9). As it stands it furnishes the 

 integral over the single interval / to t n+l . If it is desired to find the integral 

 from t n to tn+m, the formula for doing so is obviously the sum of m formulas 

 such as (9), the value of the subscript going from n + i to n + m + i, or 



In, m = 



24 



On applying this formula to the numbers of .Table I, it is found that 

 / = / sin / dt = 5[(-5 00 + -5736 + .6428 + .7071 + .7660 + .8191) 



J2S 



- \ (-0774 + -0736 + .0692 + .0643 + .0589 + .0531) 

 + (-0032 + .0038 + .0044 -f .0049 + .0054 + .0058) 



+ - (.0006 + .0006 + .0006 + .0005 + .0005 + .0004)] 



= 0.3327, 



agreeing to four places with the exact value. When a table of differences is at 

 hand covering the desired range this method involves the simplest numerical 

 operations. It must be noted, however, that some of the required differences 

 necessitate a knowledge of the value of the function for earlier values of the 

 argument than the lower limit of the integral. 



10.40 Reduced Form of the Differential Equations. Differential equations 

 which arise from physical problems usually involve second derivatives. For 

 example, the differential equation satisfied by the motion of a vibrating tuning 



fork has the form 



d?x , 



ffi - ~ kx > 



where k is a constant depending on the tuning fork. 



10.41 The differential equations for the motion of a body subject to gravity 

 and a retardation which is proportional to its velocity are 



d 2 x __ dx 

 = ~ C dt' 



where c is a constant depending on the resisting medium and the mass and shape 

 of the body, while g is the acceleration of gravity. 



