certain Differential Equations. 367 



(i>'-"-^+^).=o. 



This, when reduced, becomes 



(T^ 2^ n .n + 1 „\ 



and consequently its solution by aid of (2) is easily seen to be 



Ex. 6. The more general equation 



\ fx fx x^ a;V 



has for its solution, by (5), 



In general, if we make a' = 2r", the operation x-r- is equiva- 



z d (z d\ 



lent to —-J-, and accordingly ^(a?D) is transformed into 0( - t- j. 



This transformation leads immediately to the soluble forms of 

 Ricati's equation ; for wi-iting it in its transformed shape, viz. 



or 



(.rD . (a^D — l)-cV('-X))y=:0; 



if we assume x^-^=.z, this equation becomes 



(.D...(.D.-jl-J-(^-^)'..)y=0. 



It is readily seen, that, in order that this equation should admit 



of a finite solution, we must have =. — r- =2r + l, when r is any 



positive or negative whole number. Making this substitution, 

 we get 



(zD . (^•D-2M^)-«V)y=0, where a=(2r+l)c; 



the solution of which equation we have already seen to be (1), 



or _x_ _i_ 



= ^Tc'~^V ' ^^''■+'^''""" + A'e-(2'-+l)-^^^'], (11) 



