Example 6.  Use Mathematica to find the analytic solution and graph for the I.V.P.  [Graphics:Images/TaylorDEMod_gr_122.gif].  

Solution  6.

[Graphics:../Images/TaylorDEMod_gr_123.gif]


[Graphics:../Images/TaylorDEMod_gr_124.gif]
[Graphics:../Images/TaylorDEMod_gr_125.gif]
[Graphics:../Images/TaylorDEMod_gr_126.gif]
[Graphics:../Images/TaylorDEMod_gr_127.gif]
[Graphics:../Images/TaylorDEMod_gr_128.gif]

Dig out the formula for the solution out of the data structure of  solset and put it in  f[t].
Plot the analytic solution at the same sample points that were used for the numerical approximations.

[Graphics:../Images/TaylorDEMod_gr_129.gif]

[Graphics:../Images/TaylorDEMod_gr_130.gif]

[Graphics:../Images/TaylorDEMod_gr_131.gif]
[Graphics:../Images/TaylorDEMod_gr_132.gif]

Just for fun, plot the Taylor  series solution and the analytic solution. Notice that there is a difference.

[Graphics:../Images/TaylorDEMod_gr_133.gif]

[Graphics:../Images/TaylorDEMod_gr_134.gif]

[Graphics:../Images/TaylorDEMod_gr_135.gif]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) John H. Mathews 2004