228 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS 



10.42 The differential equations for the motion of a body moving subject to 

 the law of gravitation are 



*y~ & y - 



J-i9 ~ * ~3> 



dP r" 



r 2 = x 2 + y 2 + z 2 . 



10.43 These examples illustrate sufficiently the types of differential equations 

 which arise in practical problems. The number of the equations depends on 

 the problem and may be small or great. In the problem of three bodies there 

 are nine equations. The equations are usually not independent as is illustrated 

 in 10.42, where each equation involves all three variables x, y, and z through r. 

 On the other hand, equations 10.41 are mutually independent for the first does 

 not involve y or its derivatives and the second does not involve x or its deriva- 

 tives. The right members may involve x, y, and z as is the case in 10.42, or 

 they may involve the first derivatives, as is the case in 10.41, or they may 

 involve bo'th the coordinates and their first derivatives. In some problems 

 they also involve the independent variable t. 



10.44 Hence physical problems usually lead to differential equations which are 

 included in the form 



dx dy } 



ffiy 



dx dy 



where / and g are functions of the indicated arguments. Of course, the number 

 of equations may be greater than two. 

 10.45 If we let 



dx dy 



dt' y ~ dt' 



equations 10.44 can be written in the form 



= 

 dt ' 



dx' 



=/(#?,*'/,*), 



dy' 



dt 



, *', /, t). 



