LINEAR DIFFERENTIAL EQUATIONS 353 



suggested that comparison with Picard's method is weighted in our favor. 

 This does not seem to be the case, as will be illustrated in example 4 where 

 the exact solution is a monotonically increasing function. It has also been 

 suggested that Picard's solution might be improved by starting from a better 

 initial approximation, say Wq , rather than from the linear approximation 

 Lq , but we have not found any marked improvement in the succeeding 

 approximations (see examples 1 and 2). The various points of interest 

 will be brought out in our examples, with the accompanying figures, which 

 we shall now briefly describe. In each figure the heavy curve is the ac- 

 curate solution while the approximations are indicated by self-explanatory 

 letters. 

 Example 1, Fig. 1 



y'' = -rry, ^ X ^ 2 



y(0) = 1, y(0) = 



Exact solution: y{x) = V{l)r^'^ x'^ J-^ {Ix"") 



(a) Wave perturbation 



Wq — cos X 



Wi = -\x cos X + i(l + 2ic - x') si 



(b) Linear perturbation 



Lo = 1 



(c) Linear perturbation using initial sinusoidal approximation 



Lq = cos a: = Wq 

 Li = X -{- X Qos X — 2 sin x 

 Example 2, Figs. 2, 3 and 4 



y = -xy, 2 ^ a; ^ 6 



y(2) = 0, y\2) = 1 

 Exact solution: 



y{x) = -,2.mZx"'j.„^{ix'") - .Q\929\x"^Jx,z{ix"') 



