certain Differential Equations. 369 



in equation (1), we immediately obtain the solution already given 

 by Dr. Havgreave : 



^ \ a . «.r 2.4. (a^)^ / 



,, /, m.im + V) (m-1) . . . (to + 2) „ \ 



\ 2 .ax 2,4. (ao:)'' / 



Equations of the form 



are easily reducible to the proposed form (D^ + a^)y = 0; for if 

 we assume /^=X^, then this equation becomes 



(X«D2 + XX'D + «2)y = 0, 

 or 



((XD)2 + «2^y = 0; (12) 



hence '\i z=. I ^, the equation is reduced to the required form, 



Ex.\. ( (c2 - «2)D2 _ a;D + a2^y = 0. 



Here P dx • ,* 



z=i I — T===^5 = sm~^ - : 



J ^c^-x^ c ' 



.'. ?/ = Acos( «sin~'- +aj. . . . (13) 



Ex.2. (^{l+x'^)W + x'D + a^)y=0. 

 In this case /* Dx , , . 



and accordingly we get 



y=A(a^+ 'v/f+^)"''~'+A'(a^+ 'v/T+^)-<'^~i. . (14) 



Ex. 3. Again, let it be proposed to integrate 



(D2 - cot a?D + « sin2a')y = 0. 

 In this case 



z ^fsiu xdx = — cos a;, 



and consequently we have for the solution 



y = A cos (tf cosai' + a). ..... (15) 



III. If we assume x-=.t, y^tv, then it can be readily shown 



