Example 6.  Use Mathematica to find the analytic solution and graph for the I.V.P.  [Graphics:Images/Euler'sMethodMod_gr_134.gif].  

Solution 6.

[Graphics:../Images/Euler'sMethodMod_gr_135.gif]


[Graphics:../Images/Euler'sMethodMod_gr_136.gif]
[Graphics:../Images/Euler'sMethodMod_gr_137.gif]
[Graphics:../Images/Euler'sMethodMod_gr_138.gif]
[Graphics:../Images/Euler'sMethodMod_gr_139.gif]
[Graphics:../Images/Euler'sMethodMod_gr_140.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/Euler'sMethodMod_gr_141.gif]

[Graphics:../Images/Euler'sMethodMod_gr_142.gif]

[Graphics:../Images/Euler'sMethodMod_gr_143.gif]
[Graphics:../Images/Euler'sMethodMod_gr_144.gif]

Just for fun, plot the Euler and modified Euler solutions and the analytic solution. Notice that there is a difference.

[Graphics:../Images/Euler'sMethodMod_gr_145.gif]

[Graphics:../Images/Euler'sMethodMod_gr_146.gif]

[Graphics:../Images/Euler'sMethodMod_gr_147.gif]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) John H. Mathews 2004