Example 2.  Find the analytic solution to the second order  I.V.P.  

    [Graphics:Images/PendulumMod_gr_27.gif]  with  [Graphics:Images/PendulumMod_gr_28.gif],   [Graphics:Images/PendulumMod_gr_29.gif].    

Plot the solution over the interval
[Graphics:Images/PendulumMod_gr_30.gif].  

Solution 2.

[Graphics:../Images/PendulumMod_gr_31.gif]
[Graphics:../Images/PendulumMod_gr_32.gif] [Graphics:../Images/PendulumMod_gr_33.gif]
[Graphics:../Images/PendulumMod_gr_34.gif]

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

[Graphics:../Images/PendulumMod_gr_35.gif]

[Graphics:../Images/PendulumMod_gr_36.gif]

The graph is now plotted.  

[Graphics:../Images/PendulumMod_gr_37.gif]

[Graphics:../Images/PendulumMod_gr_38.gif]

 

 

 

 

 

 

 

 

 

 

 

 

(c) John H. Mathews 2003