54 DIFFERENTIAL AND INTEGRAL CALCULUS* 



or equating coefficients of ce r , 



jpr + Y = p therefore * - ( * * 



therefore ^ t = 7z^4 , ^4 2 = n (n - 1 ) A Q , &c., 

 and therefore 



n(n-l) , n(n-l] (n-2) , 



3 



which would satisfy the differential equation whatever 

 A n be ; but since in this case y 1 when x = 0, we have 

 A Q = 1, and obtain the expansion 



The fact observed here that the solution of this differential 

 equation involves one arbitrary constant and no more is 

 true for all differential equations of what is called the 



first order, i. e. between ~ , ?/, x. So the general solution 



of one of the second degree will involve two arbitrary 

 constants and so on, it being obvious that in the case 

 of an equation, say of the second degree, we may, for 



any proposed value of a?, give y and -j- what values we 



j>2 



choose, but when these values are chosen *[ and there- 



fore all subsequent differential coefficients are determined 

 by the equation. Thus, in the equation 



, M d*y du 

 (l-x 2 ) -5-ir-f =0, 

 1 ax dx 



suppose we choose that when ic = 0, y = a, and -r- = b, 

 and assume accordingly 



x* x r 



-+...+ -+...; 



