On the Solution of cert(nn Differential Equations. 365 



This equation is equivalent to 



( {xT) + n) {xD - ^m ) + a^v'^Jy = 0, 

 or Qv D (^D - 2n+l ) + a'^x^')xy = 0. 



Consequently its solution is by the last 



(d \" 

 J- a-^ ) . cos {ax + a) (2) 



Ex.S. (D^^^)D + «^)y=0, 



or (xJ){xI) + 2m^1) + fflV)y = ; 



or, as it can be otherwise written, 



(a; D (a; D - 2^3) + « V).r2»+ • y = 0. 

 Accordingly the solution is 



y = Ax-^"+'.(^a-'y\ cos {ax -{■»). . . (3) 



Ex. 4. If we had substituted for y, fx .u instead of x^u in 

 Ex. 2, the equation would have taken the form 



(d>-(^-<.'»«=o, 



\ JX JX X"^ / 



its solution being 



"=^-(l""0"''"'('^^+-") (^) 



If we maVc -^ = '^'x, o\'fx = e^^^, this equation is immediately 



seen to be identical with that treated by Dr. Hargreave in the 

 Philosophical Transactions for 1848, and since discussed by 

 Mr. A. H. Curtis in the Cambridge and Dublin Mathematical 

 Journal for the year 1854. 



II. In any equation if x be changed into -, the operation x-j- 



bccomes —z-j-; and accordingly, if the solution of any equa- 

 tion of the form ^{x'D)y=x he known, we can immediately de- 

 termine that of the analogous equation <l>{ — x'D)y =•)('. I will 

 illustrate the use of this method of transformation by the solu- 

 tion of a few well-known equations, and then proceed to apply 

 it to the differential equations analogous to those I have already 

 solved. 



