422 Mr. Herapath on the Integration 



General Integration of Linear Equations. 



Having laid open an easy method of proceeding from the 

 integration of equations of an inferior to those of a superior 

 order, we shall now take up the integration of the equation 



d^'y + Ad'^-'^y + Brf^-2j/ + . . . . Rj/ = X . . (7) 



on principles much more general than those we have em- 

 ployed in the preceding cases. 



It has already been remarked, and is indeed obvious enough, 

 that if r be either of the roots of 



r" + Ar"-' + Br«-2 + . . . . R = (8) 



y = e'"*is a value that will satisfy the conditions of (7) when 

 the function X = 0. Consequently, p being a function of x, 

 our business will be to determine its form so that y —jpe'^^ 

 may be the complete integral sought. This it is manifest can 

 only be done from the properties of the given equation, in a 

 manner similar to that in which we have already gone. For 

 instance, because y =pe'''^ . 

 dy = e^'irp + dp} = ^"{r + d} p 

 cPy^ e^'^r'p + 2rdp + d^p} =e^^{r + dYp 

 d?y - e^'^ir^p + ST^dp + 3rd'p + d^p} = e^'lr + dy p 



d^y = e'-Mr> + nr^-Hp + «!Lzi,.'^-Vp + . . .rf"p) = 



e'"{r + d )''p. 



Hence employing the contracted operations which relate to p 



fd^'y 



Ad^-'y 



B^-% 

 ^=icd--3y) = e"i{r + drp + A{r + dr-'p + Bir + dr-^p 



But r being a root of (8), the sum of all the first terms or 

 coefficients of p vanish. Dividing therefore by e^'^, and put- 

 ting 2„, 3„, 4.„. . . . and 2„_p 3„_i, 4^_p . . . for the bi- 

 nomial 



