Example 2.  Find the analytic solution to the second order  I.V.P.  
    [Graphics:Images/PendulumMod_gr_46.gif]  with  [Graphics:Images/PendulumMod_gr_47.gif],   [Graphics:Images/PendulumMod_gr_48.gif].    
Plot the solution over the interval
[Graphics:Images/PendulumMod_gr_49.gif].  

Solution 2.

[Graphics:../Images/PendulumMod_gr_50.gif]

[Graphics:../Images/PendulumMod_gr_51.gif]


[Graphics:../Images/PendulumMod_gr_52.gif]
[Graphics:../Images/PendulumMod_gr_53.gif]

We usually prefer our solutions in terms of real functions of a real variable.  

[Graphics:../Images/PendulumMod_gr_54.gif]
[Graphics:../Images/PendulumMod_gr_55.gif]

The graph is now plotted.  

[Graphics:../Images/PendulumMod_gr_56.gif]


[Graphics:../Images/PendulumMod_gr_57.gif]

[Graphics:../Images/PendulumMod_gr_58.gif]
[Graphics:../Images/PendulumMod_gr_59.gif]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) John H. Mathews 2004