96 Mr. G. Boole on a Class of Differential Equations. 



XX. Note 071 a Class of Differential Equations. 

 By George Boole, Esq. 



To the Editors of the Philosophical Magazine and Journal. 



Gentlemen, 



THE following may be deemed a proper supplement to 

 the remarks, offered in common by Mr. Hargreave and 

 myself, in the January Number of the Philosophical Magazine 

 on the subject of the Rev. B. Bronwin's differential equations. 

 In the greater number of those equations, the independent 

 variable x only entered in the first degree. All linear equa- 

 tions which possess this character, whether in differentials or 

 in finite differences, admit of a complete symbolical solution, 

 from which the particular solutions assigned in the papers 

 referred to, may be regarded as deductions. 



Every equation, indeed, to which this characteristic belongs, 

 may be expressed in the form 



a?<p(D)?^ + 4/(D)i^=X; (1.) 



D standing for -j-, X being a function o^l x, and cp and 4' de- 

 noting arbitrary functions or combinations of the symbol to 

 which they ai'e affixed. 



The complete solution of (1.) is 



«={<p(D)}-^^^''^a:-S-^<^'X, . . . (2.) 

 the form o^xij^) being given by the equation 



The analogy which exists between the above solution and 

 that of the linear differential equation of the first order, it is 

 scarcely necessary to notice. It belongs to a class of subjects 

 ■which have been considered in a paper on the Theory of 

 Developments, published in the Cambridge Mathematical 

 Journal, vol. iv. p. 214. 



As a particular illustration of (2.), let us suppose that the 

 given equation is of the form 



X <p(D)?< + m <p'(D)?i= X. 

 "We have 



>^^')=f- 



-^ dt = m\og<p{t\ 



<P(0 



.-. ?/= {^(D)}'»-''^-'{(p(D)} -'» X, 



