352 BELL SYSTEM TECHNICAL JOURNAL 



subdivide the interval and obtain separate approximations, though this is 

 not necessary if F is predominantly of one sign throughout (a, h). To (9) 

 we now add the sequence 



Wn{x) ={[ f{u)Wn-iiu) sin ^(x - u) du, (lO) 



P •'a 



and the series 



y{x) = W,{x) + Wr{x) + W^{x) + . . . (11) 



is the desired solution. 



The flexibility of the wave perturbation method as compared with Picard's 

 linear method lies essentially in the introduction of the variable parameter /3. 

 Since we make jS depend on the length of the interval (a, h) in which a solu- 

 tion is desired the approximations may be extended over much longer 

 intervals than is feasible in Picard's method. If F{x) is a slowly varying 

 function throughout (a, h), so that /(a:) is small, it will be found that the 

 first approximation Wq{x) is good, and the second, Wq + Wi is generally 

 adequate. 



Another choice for ^ is 



/8 = ^-i- f V-/^W dx. 



— CL Ja 



(12) 



However, the integration in (8) will often be simpler than in (12). 



Picard's method is a special case of the wave perturbation method, with 

 |8 = 0. In fact, if F{x) changes sign in (a, h), then in some cases jS as defined 

 by (8) will reduce to zero. 



If F{x) is a rapidly varying function, or if the solution is desired over an 

 infinite interval, it is usually advantageous to transform equation (1) by 

 first introducing a new independent variable 



e = j V^Hx) dx, (13) 



and then removing the first order term in the new equation by an appro- 

 priate transformation of the dependent variable. 



III. Examples 



For our illustrations we have used mainly the simple equation 



/' = -xy (14) 



whose exact solution can be expressed in terms of Bessel functions of order 

 d= 1/3. Since the Bessel functions are oscillatory in nature it might be 



