Example 7.  Given  [Graphics:Images/RungeKuttaFehlbergMod_gr_120.gif]  with  [Graphics:Images/RungeKuttaFehlbergMod_gr_121.gif]  over  [Graphics:Images/RungeKuttaFehlbergMod_gr_122.gif].  This D.E. is known to be a Riccati equation.  
The analytic solution is known to be  [Graphics:Images/RungeKuttaFehlbergMod_gr_123.gif];  compare it with the solutions that were found in Example 6.  

Solution 7.

Plot the analytic solution over the given interval.

[Graphics:../Images/RungeKuttaFehlbergMod_gr_124.gif]


[Graphics:../Images/RungeKuttaFehlbergMod_gr_125.gif]

[Graphics:../Images/RungeKuttaFehlbergMod_gr_126.gif]

[Graphics:../Images/RungeKuttaFehlbergMod_gr_127.gif]

[Graphics:../Images/RungeKuttaFehlbergMod_gr_128.gif]

[Graphics:../Images/RungeKuttaFehlbergMod_gr_129.gif]




[Graphics:../Images/RungeKuttaFehlbergMod_gr_130.gif]


[Graphics:../Images/RungeKuttaFehlbergMod_gr_131.gif]

[Graphics:../Images/RungeKuttaFehlbergMod_gr_132.gif]

[Graphics:../Images/RungeKuttaFehlbergMod_gr_133.gif]

[Graphics:../Images/RungeKuttaFehlbergMod_gr_134.gif]

[Graphics:../Images/RungeKuttaFehlbergMod_gr_135.gif]

[Graphics:../Images/RungeKuttaFehlbergMod_gr_136.gif]



[Graphics:../Images/RungeKuttaFehlbergMod_gr_137.gif]


[Graphics:../Images/RungeKuttaFehlbergMod_gr_138.gif]

[Graphics:../Images/RungeKuttaFehlbergMod_gr_139.gif]

[Graphics:../Images/RungeKuttaFehlbergMod_gr_140.gif]

[Graphics:../Images/RungeKuttaFehlbergMod_gr_141.gif]

[Graphics:../Images/RungeKuttaFehlbergMod_gr_142.gif]

[Graphics:../Images/RungeKuttaFehlbergMod_gr_143.gif]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) John H. Mathews 2004