LINEAR DIFFERENTIAL EQUATIONS 351 



tive. It will be recalled that in Picard's formulation the differential equa- 

 tion (1) is replaced by an integral equation 



y{x) = y{a) -}- {x - a)/ {a) + f F{u)y{u){x - u) du (3) 



•'rt 



where y{a) and y'{a) are assigned initial values*. Writing 



Lo{x) = y{a) + {x - a)y{a), (4) 



Lnix) = I F{u)Ln-i{u){x — u) du, n = 1,2,3, '" , 



the series 



y{x) = Lo{x) + L,(x) + L2(x) + • • • (5) 



is shown to converge to a solution of the original equation. In practical 

 applications, unfortunately, it is usually found that the successive approxi- 

 mations converge rather slowly unless the interval (a, b) is small. 



In the wave perturbation method we first rewrite equation (1) in the 

 form 



y- = -^'y + 1^2 + F{x)]y = -^y + f{x)y, (6) 



and instead of the integral equation (3) we use 



y{x) = y{a) cos ^{x " ^) + B >''(^) sin ^{x - a) 

 -\- - I fMyM sin ^{x — u) du. 



(7) 



The parameter ^ is arbitrary and might be defined in various ways. We 

 have found it convenient to use the definition 



0'= - f^ ( Fix) dx, (8) 



— a Ja 



so that if F{x) is negative /3 is real and our first approximation 



Woix) = y{a) cos ^{x - «) + ^ y'M sin ^{x - a) (9) 



is sinusoidal. If F is positive /3 is imaginary and we start with an exponen- 

 tial approximation. If F changes sign in (a, b) the best procedure is to 



* This is not quite the usual form of the integral equation but it is substantially 

 equivalent. 



