300 ON THE SOLUTION OF THE DIFFERENTIAL RESOLVENT. 



Professor Boole^s * Treatise on Differential Equations/ 

 page 427. 



'^ If a linear differential equation^ whose second member 

 is zero, be reduced to the symbolical form 



then a particular solution will be u=.Xume^^, the value of 

 the index (m) in the first term being any root of the equa- 

 tion f^{m) = o, the corresponding value of u^ an arbitrary 

 constant, and the law of the succeeding constants being 

 expressed by the equation 



This rule is proved immediately by substituting the 

 series as the value of {u) in the above differential equation. 

 Let us apply this to the quartic resolvent 



(D-Z)(„_L»)(,-a) __ 



y D(D-i) (D~2) ^ y~^' 



derived from the algebraical equation y^—^y-\-'^x'=o. 

 The equation in (m) given above becomes 



m{m—i) [m — %)Urrv—yri—-j\m jim ^\um-^ = o, 



and the equation /o(m) = o becomes m{m—i) (m — 2) =0. 



Taking the root m = 2 as the initial value, we determine 

 the coefficients of the series in succession by putting 



m = 5, 8, II w. . . 



and we have 



13 10 7 

 5.4.3. u,= — . — . ^u^ 



^ "^ ' 4 4 4 



o ^ 25 22 IQ 



^ '4 4 4 ' 



1 



•••«5 = 



13 IP _ 



4*4*4. 



