

NUMERICAL SOLUTION OF DIFFERENTIAL EQUATIONS 



221 



then / = F(b) - F(a). But suppose no F(x) can be found satisfying (2). It 

 is nevertheless possible to prove that the integral / exists, and if the value of 

 (x) is given for every value of x in the interval a ^ x ^ b, it is possible to find the 

 numerical value of 7 with any desired degree of approximation. That is, it is 

 not necessary that the primitive of the integrand of a definite integral be known 

 in order to prove the existence of the integral, or even to find its value in any 

 particular example. 



10.03 The facts are analogous in the case of differential equations. Those 

 having numerical coefficients and prescribed initial conditions can be solved 

 regardless of whether or not their variables can be separated. They need to 

 satisfy only mild conditions which are always fulfilled in physical problems. 

 It is with a sense of relief that one finds he can solve, numerically, any particular 

 problem which can be expressed in terms of differential equations. 



10.04 This chapter will contain an account of a method of solving ordinary 

 differential equations which is applicable to a broad class including all those 

 which arise in physical problems. A large amount of experience has shown that 

 the method is very convenient in practice. It must be understood that there is 

 for it an underlying logical basis, involving refinements of modern analysis, 

 which fully justifies the procedure. In other words, it can be proved that the 

 process is capable of furnishing the solution with any desired degree of accuracy. 

 The proofs of these facts belong to the domain of pure analysis and will not be 

 given here. 



10.10 Simpson's Method of Computing Definite Integrals. The method of 

 solving differential equations which will be given later involves the computation 

 of definite integrals by a special process which will be developed in this and the 

 following sections. 



Let t be the variable of inte- 

 gration, and consider the definite 

 integral 



i. F = 



This integral can be interpreted 

 as the area between the /-axis and 

 the curve y = /(/) and bounded 

 by the ordinates t = a and t = b, 

 figure i. 



Let to = a, t n = b, yi = f(ti), and 

 divide the interval a ^ K b up into 

 n equal parts, each of length h = 

 (b - a)/n. Then an approximate value of F is 



FIG. i 



2. 



F = 



This is the sum of rectangles whose ordinates, figure i, are y 1} y^ . . . , y n . 

 10.11 A more nearly exact value can be obtained for the first two intervals, 

 for example, by putting a curve of the second degree through the three points 



