DIFFERENTIAL AND INTEGRAL CALCULUS. 53 



equation as before, except that in the former case the 

 equation is true when n 0, which is not so here. 



The two functions sin (m sin' 1 ^), and cos(m sin" 1 ^), each 

 satisfy the equation 



and therefore also the equation 



... <F +t2 y cF +l y .. 



( l - x) dx n "- (2n +V X ^ +(m n 



The function e fflsm " laj satisfies the equation 



v ; dx* dx 



(It will be observed that this may be deduced from the 

 last by putting a V(- 1) for % an( i therefore the resulting 

 equation connecting the higher differential coefficients i& 



So in general functions of any form may be eliminated 

 by differentiation, and a relation found connecting their 

 differential coefficients. Such a relation is called a dif- 

 ferential equation. The principal use made of them in 

 the differential calculus is to find the law connecting the 

 successive coefficients of different powers of x when the 

 function is expanded into a series of integral powers of ar. 

 Thus, if we had not the Binomial Theorem, and found 

 the differential coefficient of (1 + x) n to be n (1 -f x} n ~ l other- 

 wise (which is not difficult), we should have if y = (1 + x)\ 



the differential equation (1 + x) -jj- = ny -, and assuming y 

 to be 



then (l+x)(A l + Aj e +...+ A r ^- I + A rH ^- r +...) 



L*L 



