106 INTEGRATION OF DIFFERENTIAL EQUATIONS. 

 and, as except when n = 1, there is 



nb n + gb^ = 0, ^-~ = - - b^ l9 unless n = 0, 



a 



.-. . = &. + -& ........................ (a'). 



Now the solution of the auxiliary equation is 



z = c 1 + c z e~* x ; 

 and from (a') we deduce 



v z -\ -- z : 

 qx 



therefore y = (l + } (c t + W**) 



\ 2*E/ 



is apparently the solution of the proposed equation. But it will 

 be found not to satisfy it, unless c t = 0, and then 



is only a particular solution. The reason is, that in laying down 

 (a') as generally true, we imply that 



nb n + gb^ = 



is true for n = 1 ; whereas the equation which contains tlxe 



, ,. d*z dz 

 solution of -j-5 + q -y- = 0, viz. 

 dx* s dx 



shows that b t is not necessarily connected with b ; and that if 

 we assume such connection, we get only a particular solution. 

 Hence our formula of reduction implies the connection of b^ and 

 b ; while their independence is implied in the general solution 

 of the auxiliary equation, to which this formula is applied j and 

 these contradictory suppositions lead to an erroneous result. 

 To put ^ = 0, is to connect b Q and b lt or, which is the same 

 thing, to neglect the factor n - 1 ; and the value of y thus got 

 is therefore a solution, but not the complete solution of the pro- 

 posed equation. 



To complete it, we must, bearing in mind the independence 

 of 5 , recur to (a), which is always true, 



