Example 8.  Use the extended Runge-Kutta method to compute a numerical approximation for  

            [Graphics:Images/PainlevePropertyMod_gr_221.gif]   with   [Graphics:Images/PainlevePropertyMod_gr_222.gif]   over the interval  [Graphics:Images/PainlevePropertyMod_gr_223.gif].  

Solution 8.

    The companion differential equation is  [Graphics:../Images/PainlevePropertyMod_gr_224.gif]  where   [Graphics:../Images/PainlevePropertyMod_gr_225.gif]. 

[Graphics:../Images/PainlevePropertyMod_gr_226.gif]

[Graphics:../Images/PainlevePropertyMod_gr_227.gif]

 

 

[Graphics:../Images/PainlevePropertyMod_gr_228.gif]

[Graphics:../Images/PainlevePropertyMod_gr_229.gif]

[Graphics:../Images/PainlevePropertyMod_gr_230.gif]

 

 

[Graphics:../Images/PainlevePropertyMod_gr_231.gif]

[Graphics:../Images/PainlevePropertyMod_gr_232.gif]

 

 

[Graphics:../Images/PainlevePropertyMod_gr_233.gif]

[Graphics:../Images/PainlevePropertyMod_gr_234.gif]

[Graphics:../Images/PainlevePropertyMod_gr_235.gif]

 

 

[Graphics:../Images/PainlevePropertyMod_gr_236.gif]

[Graphics:../Images/PainlevePropertyMod_gr_237.gif]

[Graphics:../Images/PainlevePropertyMod_gr_238.gif]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) John H. Mathews 2005