10 REPORT 1861. 



On Petzval's Asym];)totic Method of solving Differential Equations. 

 By Wllliam Spottiswoode, M.A., F.R.S. 



The researches of M. Petzval here brought under notice are directed to the sohi- 

 tion of those linear differential equations with variable coefficients which have 

 reference to motions, themselves small, but propagated to gi-eat distances. In such 

 equations y usually represents the distiu'bance, and x the distance from the origin. 

 If then the solution y=:f(x) be considered as the equation to a cm-ve, the method 

 proposed by the author will give the values of y con-espouding to large values of x ; 

 in other words, the asymptotes to the curve in question. Hence the name " Asym- 

 ptotic Solution." 



With a view to this object M. Petzval proposes the following question : Can any 

 general laws be established, with respect to the coefficients of a differential equa- 

 tion, capable of furnishing criteria for determining the natm'e of the particular in- 

 tegrals which satisfy it? Having first made such a classification of functions as 

 renders his conclusions capable of conversion, in the logical sense of the term, he pro- 

 ceeds to form, from a given equation of the degi-ee n, 



X„y(»)+X„_i2/(»-i)+ . . Xoy = (X,y)(»)=0, 



the equation of the degree («+»•), (Z,s)(n+''")=0, arising from the introduction of r 

 particular integrals of a given form. 



Passing over the case of algebraic integrals, some of the criteria of which are com- 

 mon to exponentials, the more important cases are as follow : — 



I. Particular integrals of the form €«-«^Q, where Q is an entire algebraic poly- 

 nomial. 



(1) To a level (i. e. an equality of degrees among consecutive coefficients) in 

 (X, «/)(») =0, there con-esponds in general a level among those of 



(Z,z)(»+i)=0. 



(2) To a level among Xj+r-i, Xk+r-2, ■ • Xj, followed by a continuous fall 

 among Xjt-i, Xyt_2, . . Xo, of (X, ?/)(«) =0, there corresponds a level among 

 Zk+r, Zk+r-1, . . Zi, followed by a similar faU among Zk-i, Zk-2, . ■ Zo, of 



(Z,£)('^+1) = 0. 



H. Of the foi-m e'-^'+'/'C*) Q, or eP^ Q, where ^(x) is defined to belong to the 

 author's first class. 



(1) To a continuous rise among Xn, X»-i, . . Xn-r+i, of (X, i/)(»)=0, there 

 con-esponds in general a similar rise among Z,„ Z„— i, . . Zn-r, of (Z,s)('"+i) =0. 



(2) To a continuous fall among Xj-i, Xt_2, . . Xo, of (X,y)('')=0, there cor- 

 responds in general a similar fall among Zi_i, Zi_2, . . Zo, of (Z, z)('«+i)=0. 



HI. Of the form ef <fi.x)dx (^^ where the degree o{/(j>(x)dx is fractional, =•?, and 



consequently that of ^(x) is ^ — ?= — -. 



9 ? 



(1) If " be a proper positive fraction, to a level among Xk-i, Xjc-2, ■ • Xo, 



of(X,2/)(«)=0,therecoiTespondsafallamongZ^_i,Zr-2,..Zo,of (Z,3)('>+'-)=0, 

 amounting in aU to — . 



(2) If ^ be an improper positive fi-action, to a level among Xb, Xn-i, . . 



Xn~r+i, of (X, ?/)(«)=0, there corresponds a rise among Zn+r, Zn+r-i, . - 

 Z„+i, of (Z,s)(''+'-)=0. 



n r '^^"^ d x 



IV. Of the forms e'/'W _!* , and J C*-")"* ' 



to a series of coefficients X„, Xn-i, . . Xn^r+l, of (X, y)(«)=0, free 

 fi'om the factor {x — a), there coiTesponds a series (a; — a) «+*+•• Zn+r, 

 (x-a)»+" Z„+r-l, . . (x-ayZn+l, of (Z,z)>^+r=0. 

 The general result to which the author brings these conclusions, together with 



the exceptional cases, not here specified, wiU be best exhibited by the following 



examples. 



