X. NUMERICAL SOLUTION OF 

 DIFFERENTIAL EQUATIONS 



BY F. R. MOULTON, PH.D., 

 Professor of Astronomy, University of Chicago; 

 Research Associate of the Carnegie Institution of Washington. 



INTRODUCTION 



Differential equations are usually first encountered in the final chapter of 

 a book on integral calculus. The methods which are there given for solving 

 them are essentially the same as those employed in the calculus. Similar methods 

 are used in the first special work on the subject. That is, numerous types of 

 differential equations are given in which the variables can be separated by 

 suitable devices; little or nothing is said about the existence of solutions of 

 other types, or about methods of finding the solutions. The false impression 

 is often left that only exceptionally can differential equations be solved. What- 

 ever satisfaction there may be in learning that some problems in geometry and 

 physics lead to standard forms of differential equations is more than counter- 

 balanced by the discovery that most practical problems do not lead to such 

 forms. 



10.01 The point of view adopted here and the methods which are developed 

 can be best understood by considering first some simpler and better known 

 mathematical theories. Suppose 



i. F(x) = x n + a& n - 1 + ..... + a^-ix + a n = o 



is a polynomial equation in x having real coefficients ai, a?, . . . , a n . If n is 

 i, 2, 3, or 4 the values of x which satisfy the equation can be expressed as explicit 

 functions of the coefficients. If n is greater than 4, formulas for the solution 

 can not in general be written down. Nevertheless, it is possible to prove that n 

 solutions exist and that at least one of them is real if n is odd. If the coefficients 

 are given numbers, there are straightforward, though somewhat laborious, 

 methods of finding the solutions. That is, even though general formulas for 

 the solutions are not known, yet it is possible both to prove the existence of the 

 solutions and also to find them in any special numerical case. 



10.02 Consider as another illustration the definite integral 



i. /= 



where f(x) is continuous for a ^ x ^ b. If F(x) is such a function that 



-* 







220 



