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XLV. On the Solution of ceiiain Differential Equations. 

 By Benjamin Williamson, Fellow of Trinity College, Dublin'^. 



IN " A General Method in Analysis/' published in the Trans- 

 actions of the Royal Society for the year 1844, Professor 

 Boole proposes a method for the reduction of differential equa- 

 tions to others already soluble, and gives several examples of its 

 application. 



The object of the present paper is to exhibit some of Professor 

 Boole's results in another form, to apply the same method to 

 another class of differential equations, and to extend such solu- 

 tions to certain analogous pai'tial differential equations. In 

 doing so, I will restrict myself to the consideration of the dif- 

 ferent classes of equations which depend for their solution on 

 [W±a^)y = 0. 



I. I will commence with the consideration of the equation 



(D^-^D+«^)y=0, 



where D stands for -j-. This equation is at once transformed 



into 



(xTf . {xJ) - 27+\ ) + a^'^yj = 0. 



Assume, in accordance with Dr. Boole's method, 



y = {xJ)-l) . (a?D— 3) .... (a;D-2^r^l)j/'; 

 then, since 



y{xD - 1) {xB- 2^i^^)y' = {xD - 3) . . . (o^D -2nTl)a^V. 



the proposed equation becomes 



(xD-S) {xD-2i^l){x'D . xB-l . +a^x^)y' = 0, 



or (D^ + a^)?/' = 0, the solution of which is y' = Ccos{ax + a.). 

 Accordingly that of the proposed equation is 



y = c.{x'D + 2n—l) .. . {xD — 3){x'D — ^) cos {ax + u) 



[since («^-m) = «'»(«^)..-"']. 

 E^.2. Let (p._n.n + l^^,j^^Q^ 



* Communicated by the Author. 



