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LXX. 071 the Integration of Linear Differential Equations 

 having Constant Coefficients ajid last Tervi any Function of the 

 Indeterminate Quantity x. By John Herapath, Esq. 



Definition. — T70R the sake of brevity I shall constantly in 



■*- the following researches put rf^ = 1, which 



will of course not appear in any of the differential expressions. 



It is not intended here to treat of the manner of integrating 



differential equations of the form 



—^ + ^-r^ + . . . R -^ + Sj/ = 

 dx dx dx -^ 



or, as it is proposed to write them, 



d'^y + Arf"-V + ^dy + Sj/ = (1) 



in which A, B, . . . R, S are constant quantities. One pro- 

 perty of these equations which we shall find serviceable in our 

 future inquiries is obvious enough ; namely, \\\Siipe^^ and its 

 differentials, being everywhere substituted for y and its dif- 

 ferentials, will satisfy the conditions of (1), \vhen ^ is a con- 

 stant and r a root of the equivalent algebraic equation 



r" + Ar"-^ + Br"-^ + . . . Rr + S = o (2) 



From this simple property we may at once advance to the 

 integration of 



d^'y + Arf"- V + Sj/ = X (3) 



in which X is any function of x ; but as it might appear some- 

 what abrupt to launch forth so suddenly into all the difficul- 

 ties and generality of the subject, I propose to begin with the 

 integration of two or three of the inferior cases. 



Suppose we had to integrate the equation of the first order 



dy + Ay = X. 

 If in this equation X = the iiitegral is evidently 

 y = c(?~-'^^ 

 c being an arbitrary constant. Now putting for the arbitrary 

 constant a function j^ of x, it is clear that whatever be the 

 value or form of y, the indeterminate expression yj c~ ^ must 

 contain it. The only difficulty therefore lies in determining 

 J) so as to satisfy the conditions of the proposed differential. 

 For y and its differentials let us substitute their equivalents 

 from^ = pe~ , wliich produce 



3H2 Ay 



