Wilder'' s Algebraic Solution. 271 



Art. XI. — Algebraic Solution ; by Mr. C. Wilder, of New 

 Orleans. 



Remarks on the determination of y, in y"+ai/~'' +6?/"~2 

 +c?/"-=»+^«/"-* . . . ■\-hj+l=0. 



x^+S, (A) 



If we assume the function ^ , "i t^t^ and then deter- 

 mine Sg, so that (B) may be a factor of (A), independently 

 of X, we shall have 



x-{-y ' 

 by writing y-\-z for y, it is changed to 

 ^2 _j^2 —<2zy—z^ 



x-\-y + z 

 which gives to (A) the form of the function 



tf--\-ay-\-b=0. 

 Now if we make (B)=0, we shall also have (A)=0; hence, 

 by comparison, 



2z=:a, (1), 

 z^-x^=^b, (2), these two equations 

 together with x-\-y-\-z=^0, (B), are sufficient to deter- 

 mine y ; for from ( 1 ) and (2), 



x^=a^-Ab 



a 

 and from (1) and (B),2/= — - -a:, which is the common rule. 



In like manner, assuming the function 



x^ -^-yx-^p (D)' 

 we shall have, when (D) is a factor of (C) independently of.?;, 



x^ -\-{y^ — Spy)x^ -\-p^ 



x^ -{-yx-\-p 

 writing y-{-z for ^/, and it is changed to 



x^ -\-{y^ + Szy^ -{-{Sz^ ■-^p)y+z^ -3pz)x^ +p^ 

 x^ -\-yx-\-p 

 Now when (D)=0, we also have (C)=0, and we may there- 

 fore, after dividing (C) by a^^, compare it with the equation 



y^ -\-ay^ -\-by-i-c=0, 

 which gives 3z=a, (1) 



3z^—3p=b, (2) 



