of Linear Differential Equations. 421 



for the complete integral after restoring the value of X„ each 

 vinculum { being understood to include all the quantities on 

 the right of it, and c, c, the two arbitrary constants of integra- 

 tion. 



Let us now take for a third example the integration of 



dPy + A.(Py + 'Qdy + Cj/ = X. 

 Then, as before, since every value?/ could have must be com- 

 prehended in j3 f *■ * in which r is a root of the equivalent alge- 

 braic equation, 



Cy^Cpe^^ V 



^dy ^Bipre''^ + e^'^dp} \ 



kd^y = A {pr-'e'-^ + 2re''^dp + e^'^d'p} j'=X; 



d'y = pt^e"-' + Sr^e^'^dp + 3re''^d'~p + e^'^d^p] 



or by taking the sum and considering that r is such a quantity 

 as makes the sum of the first vertical column of the develop- 

 ment vanish 



(B + 2Ar + 3r-) e^'dp + (A + 3 r) e^'^d'p + e^^d'p = X. 



Dividing this by e '' ^ and integrating, we get 

 (B + 2 A r + fj 7--)p + (A + 3 r) dp + d^p = c +/X e "'"^ 

 or B^p + A^dp + d-p = X,, 



putting A, , Bi for the coefficients of p, dp, and Xj for the 

 right-hand member, which is a function of a^. This equation 

 being a linear differential of the second order with respect to 

 p its integral is obtained from the preceding example ; for in- 

 stance it is 



p = e-(^' + '-'>{c, + y>(A,+2r.> 1^^ ^y x^ ^-r,x^ 



where r^ is a root of a quadratic whose coefficients are, 1, A„ 

 and B,. Multiplying this value by ^'"'^ and restoring the 

 values of A,, B,, and X, there results 



(6) 



the integral or value of y complete with all the arbitrary con- 

 stants required by the successive integrations. We shall pre- 

 sently examine the truth of these solutions and investigate the 

 law connecting the variable exponents as well as some other 

 properties of the solutions. 



' General 



