191 1.] LINEAR DIFFERENTIAL EQUATIONS. 275 



(c) In this identit}- arranged according to the ascending powers 

 of S equate to zero the coefficients of the different powers of S. 



(d) Solve the dift'erential equations thus obtained in regular order 

 for Vo.Vi,3'o,3'3, V4. •■•• 



(e) Substitute these values in ( 3 ) and make 5" unity. The result- 

 ing value of _v, if it contains a finite number of terms or if it is a 

 uniformly convergent infinite series, is a solution of the given dif- 

 ferential equation.^ 



The method of solution of linear dift'erential equations as here 

 outlined does not seem to occur in mathematical literature except as 

 developed by the author. 



The method will be exemplified by applying it to two dift'erential 

 equations, important in mathematical physics — Bessel's equation, a 

 second order ordinary dift'erential equation, and Fourier's equation 

 for the flow of heat, a second order partial dift'erential equation. 



Bessel's equation is 



X' 





Replace Bessel's equation by 

 and assume that 



y = y\> + y\S + y,5-^' + v..^' + y,S' + • • • 



makes the latter equation an identity. 



When arranged in ascending powers of 5^ this identity is 



o^j'o ' ,^ ^ , .^_y^ 5^' -f • • • = o. 



dx^ , "^ dx' "^ dx~ 



'^O ^1 ^2 



dx dx dx 



^ This method gives a formal solution of non-linear differential equations, 

 but up to the present time the author has been unable to test the resulting 

 series for convergency. 



PROC. AMER. PHH,. SOC. L. I99 R, PP INTED JUNE 30, I9II. 



