of Linear Differential Equations. 23 



to satisfy each, the equation proposed deprived of its last term 

 X they separately vanish, and leave 



d?y x dp + d 2 y 2 dp x + d 2 y 3 dp 2 = X which, together with 

 dy x dp + dy 2 dp x + dy 3 dp 2 = 

 and Vidp+y 2 dp t + y 3 dp 2 = 



furnishes three several equations from which two of the quan- 

 tities dp, dp x , dp 2 may be eliminated, and the value of the 

 third p 9 p t , p 2 be obtained by a simple integration. And this 

 performed with the others will afford us the values of p, p t9 

 p 2 ; and of course, according to Lagrange, the complete solu- 

 tion of the differential equation. 



In order to investigate this with every possible regard to 

 generality, let as before r, r, , r 2 be the roots of the equivalent 

 algebraic equation 



r 3 + Ar 2 + Br + C = 0; 

 and let y x = e rx , y 2 = e r,x , y 3 = e r%x 



which give dy x = re rx , dy 2 m r x e riX 9 dy 3 = r i e r2X 

 and d 2 y l = r 2 e rx , d 2 y 2 = r 2 e r < x , d% = r*e r * x 



Hence the three equations give 



r 2 e rx dp + r*e r **dp t + r 2 e r * x dp 2 = X 



re rx dp + r 1 e riX dp l + r 2 e r ' 2X dp 2 = 



e rx dp + e r > x d Pl + e r * x dp 2 = 

 from which eliminating dp 2 we obtain 



(f-rr*) e rx dp + (r?-r x r 2 ) e r * x dp, = X 

 {r -r 2)e r*dp +(r l -r 1 )e r * x dp l = 

 and then eliminating dp x from these 



(r 2 —rr 2 —rr x + r x r 2 )e rx dp = X 



c+/Xe- rx 



or P == • 



r r^—rr 2 —rr l -\-r l r 2 



In the same way the other coefficients p l9 p 2 are found; but 

 as this one will answer our present purpose, we shall not trou- 

 ble ourselves to determine them. From this coefficient we 

 obtain, 



py x = ace rx + ae rx f^e~ rx 

 putting a for the numeral reciprocal. This is one value that 

 must satisfy the proposed equation. Now in this value r 



being 



