232 The Ohio Journal of Science [Vol. XVI, No. 6, 



When ( (D — a)-+b-)" is a factor of the first member, the 

 (n — 1) partial derivatives of e'"^ with respect to m, 



ypinx v2pmx -y" 1 pnix 



are also solutions, when m is a + bi, or a — bi, giving also as 

 solutions, 



xc'^sin bx, xe^'^cos bx, x^e'^^'sin bx, 



X2gaxcQg ]3x^ ^ j^n-lgax g^j^ ^x, 



X^-l gax cos bx. 



Multiply each by an arbitrary constant and add. 



Case II. 

 For the homogeneous linear differential equation of the form, 



y = x'" 

 gives X" F(m) =0, and if m — a is a factor of F(m), a solution is 



y = x'"- 

 The partial derviatives of x", with respect to m, are 



X'" logeX; X"' log-eX; x"' log^x, etc. 

 Thus if (m — a)" is a factor of F(m), the solutions are, 



y = x% y = x'^ logeX, y = x^(logex)2, 



y = x^(logex)"-i 



Multiply each by an arbitrary constant and add. 



Case III. 



(a) In (a) of Case I, if the second member is e"'"" (instead of 

 zero) the particular solution is 



pmx pmx 



D — a m — a 



mx 



for the equation (D — a) y = e 

 This fails when m = a. 

 All failing cases of this sort will give a solution when treated 



like the form in calculus (differentiating as many times as the 



factor occurs, omitting differentiation as to factors giving no 

 trouble, just as in the calculus problem). 



