ON THE SOLUTION OF LINEAR DIFFERENTIAL EQUA- 

 TIONS OF SUCCESSIVE APPRONIMATIONS. 



Bv PRESTON A. LAMBERT. 

 (Read April 30, 191 1.) 



The object of this paper is to apply to the solution of linear differ- 

 ential equations, both ordinary and partial, the method of expansion 

 into series used in the solution of algebraic equations in the papers 

 read by the author before the Philosophical Society in April, 1903, 

 and in April, 1908. 



Let the given differential equation be 



f dy d'y (t\v\ 



o. 



The method of solution consists of the followirg steps : 

 (a) Break up the left-hand member of the dift'erential equation 

 into two parts. 



and 



/ dy d\v d"y\ 



-^\''''^''dv'dP^ ■'■'dr^) 



j\^^y^ dx • dx-' '"' dx")' 



such that the first part equated to zero can be integrated by some 

 known method, and multiply the second part by a parameter 5", inde- 

 pendent of .r and 3'. Replace the given equation by 



, . rf dy d^y d"y\ ( dy d'y d"y\ 



(6) Assume that 



(3) 3' = Vo + y\S + r.^-^' + yS-^ + r A* + • • • 



makes equation (2) an identity. 



274 



