266 ELIMINATION OF CONSTANTS. 



242. In general, if we have an equation between x and y 

 and n arbitrary constants, and we differentiate m times suc- 

 cessively, we have m + 1 equations between which we can 

 eliminate m constants, and this will give a result involving 



r- and inferior differential coefficients of y. There will 

 dx 



also be n m constants in the resulting equation ; and as we 

 can choose at pleasure the m constants we eliminate, we can 

 form as many resulting equations containing n m constants, 

 as the number of combinations that can be formed out of 

 n things taken m at a time ; that is, 



n (n 1) ... (n m + 1) 



~w~ 



Each of these resulting equations is called a differential 



d m y 

 equation of the m th order, - being the highest differential 



coefficient of y which occurs in it. 



When the original equation is differentiated n times suc- 

 ce'ssively, we have n + 1 equations, between which all the 

 constants can be eliminated, giving us a differential equation 

 of the w tb order. 



243. If we recur to the example in Art. 241, we have 

 for one of the three differential equations of the first order, 



If we find a from this equation in terms of x, y, b, and 

 ~ , and substitute in the given equation, we obtain another 



G3s 



differential equation of the first order. If we find b in terms 

 of x, y, a, and -~- , and substitute in the given equation, we 

 obtain the remaining differential equation of the first order. 



The three differential equations of the second order which 

 can be obtained by combining equations (1), (2), and (3) of 

 Art. 241, are 



