LINEAR DIFFERENTIAL EQUATIONS 



For values of x less than 1 W^, was evaluated numerically. 

 Example 6, Table II 



363 



y"^\^' = i^x^-^^^ 1^:.^3. 



:y(i) = 1, 



y(i) =0. 



The solution of this equation using Picard's method and the integraph 

 has been described by Thornton C. Fry.^ We compare his results with those 

 obtained by the wave perturbation method. The equation is first reduced 

 to normal form by the substitution y = oc~^'^ u, so that 



"" = (-^+A)~ 



and we have ^ = J\/3. Then 



x"' Wo = cos iS (x - 1) + ^ sin ^(x - 1) 



2^ 



1/2 



'^■' = if(^^-Oh^^"-^^ 



+ ^ sin j8(« — 1) sin 0(x — u) du. 



While Wi may be evaluated in terms of Ci and Si functions the values 

 tabulated below were obtained by numerical integration. The values of 

 the accurate solution 



y = 1.4034 /i(x) - 0.3251 N^{x), 



and of the third and eighth Picard approximations, are copied from Fry's 

 paper. 



Table II 



