•Ax^^-A.r^;^^ I -X, 



420 Mr. Herapath on the Integration 



Ay = Ap e~ "^^ 

 dy — — Ape' 

 or, e~^^dp = X. 



Dividing by e~ ^* and integrating, there results 



p =z c + TXe-^^ 

 and consequently 



y=pe-A^ = e-^'{c +/Xe^^} (4) 



the complete integral sought, c being the arbitrary constant. 



A first glance would lead one to suppose that this process 

 is virtually the same as Lagrange's ; but in the two subse- 

 quent cases the difference will be obvious enough. 



Given the differential of the second order 



d^y + Ady + By = K. 

 Pursuing the same train of reasoning we demonstrate that the 

 complete integral or value of j/ must be contained in pe^^ 

 assuming p to be an indeterminate function of x, and r a root 

 of the equivalent algebraic equation 



r- + Ar + B = 0. 

 Substituting therefore for y, &c. their values we obtain 



By=Bpe'-' "j 



Ady = Arpe'^ + Ae'^^dp { ~ ^ 



dhj = r^pe''''- + "i-rerx ^p + ^rx dip J 

 which on account of the value of r gives 



(A + 2r)e''^dp + e''^d^p = X. 

 Whence by division and integration 



(A + 2r-)p + dp = c+/*Xe-", 



and if the right hand member be called X, we have by the 

 preceding example 



Therefore multiplying by e'"^ we have 



or 3, = ,-(A + '-^^-J,, +^,(A+20. J,+y>X.-" (5) 



for 



