Example 5.  Form the desired solution to Example 1.

Solution 5.

Finally, the desired solution x(t) is the linear combination  

     [Graphics:../Images/ShootingMod_gr_125.gif].   

For this example we have  [Graphics:../Images/ShootingMod_gr_126.gif]  and we can locate  [Graphics:../Images/ShootingMod_gr_127.gif] in  [Graphics:../Images/ShootingMod_gr_128.gif]  and  [Graphics:../Images/ShootingMod_gr_129.gif] in  [Graphics:../Images/ShootingMod_gr_130.gif].  
Since the desired solution is  [Graphics:../Images/ShootingMod_gr_131.gif],  these ordinates are a linear combination of the two previously computed solutions.

The points (t, x(t)) are now formed.

 

[Graphics:../Images/ShootingMod_gr_132.gif]

Now we can plot the solution.

[Graphics:../Images/ShootingMod_gr_133.gif]


[Graphics:../Images/ShootingMod_gr_134.gif]

[Graphics:../Images/ShootingMod_gr_135.gif]

[Graphics:../Images/ShootingMod_gr_136.gif]

[Graphics:../Images/ShootingMod_gr_137.gif]
[Graphics:../Images/ShootingMod_gr_138.gif]


The first point is

[Graphics:../Images/ShootingMod_gr_139.gif]

[Graphics:../Images/ShootingMod_gr_140.gif]
[Graphics:../Images/ShootingMod_gr_141.gif]

The last point is

[Graphics:../Images/ShootingMod_gr_142.gif]

[Graphics:../Images/ShootingMod_gr_143.gif]
[Graphics:../Images/ShootingMod_gr_144.gif]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) John H. Mathews 2004